data_source stringclasses 1 value | prompt stringlengths 1.37k 1.43k | ability stringclasses 1 value | reward_model stringlengths 52 75 | extra_info stringlengths 117 170 |
|---|---|---|---|---|
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [69, 91, 47, 74, 75, 94, 22, 43, 50]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [22, 43, 75, 47, 69, 74, 91, 50, 94]} | {"rule_name": "palindrome_distance", "elements": [69, 91, 47, 74, 75, 94, 22, 43, 50], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [40, 51, 90, 27, 98, 85, 47]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [40, 85, 47, 90, 51, 98, 27]} | {"rule_name": "modulo_chain", "elements": [40, 51, 90, 27, 98, 85, 47], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [82, 52, 9, 65, 90, 86, 45, 95]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [9, 90, 52, 82, 45, 65, 95, 86]} | {"rule_name": "reverse_digits", "elements": [82, 52, 9, 65, 90, 86, 45, 95], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [85, 3, 46, 77, 16, 59, 32]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [32, 3, 46, 16, 77, 85, 59]} | {"rule_name": "digit_operations", "elements": [85, 3, 46, 77, 16, 59, 32], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [87, 89, 78, 59, 34, 26, 71]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [87, 89, 26, 34, 59, 71, 78]} | {"rule_name": "modulo_chain", "elements": [87, 89, 78, 59, 34, 26, 71], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [68, 80, 74, 6, 10, 53, 84, 7, 87, 3]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 6, 7, 10, 87, 53, 68, 74, 80, 84]} | {"rule_name": "palindrome_distance", "elements": [68, 80, 74, 6, 10, 53, 84, 7, 87, 3], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [26, 14, 37, 58, 96, 73]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [14, 37, 96, 26, 73, 58]} | {"rule_name": "bit_operations", "elements": [26, 14, 37, 58, 96, 73], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [43, 56, 74, 30, 71, 6, 72, 93, 83]\n\nSorting Rule: Elements ordered by proximity to Fibonacci sequence\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [6, 56, 30, 93, 83, 43, 74, 71, 72]} | {"rule_name": "fibonacci_proximity", "elements": [43, 56, 74, 30, 71, 6, 72, 93, 83], "rule_description": "Elements ordered by proximity to Fibonacci sequence"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [79, 24, 68, 85, 7, 96]\n\nSorting Rule: Elements ordered by hidden zone-based rules\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [7, 24, 68, 96, 85, 79]} | {"rule_name": "alternating_zones", "elements": [79, 24, 68, 85, 7, 96], "rule_description": "Elements ordered by hidden zone-based rules"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [95, 75, 59, 4, 67, 63, 70, 97, 84, 94]\n\nSorting Rule: Elements ordered by a hidden circular permutation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [75, 84, 94, 95, 97, 4, 59, 63, 67, 70]} | {"rule_name": "circular_shift", "elements": [95, 75, 59, 4, 67, 63, 70, 97, 84, 94], "rule_description": "Elements ordered by a hidden circular permutation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [72, 73, 83, 33, 34, 9, 91, 17, 8, 22]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [83, 22, 91, 17, 34, 72, 8, 73, 9, 33]} | {"rule_name": "bit_operations", "elements": [72, 73, 83, 33, 34, 9, 91, 17, 8, 22], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [25, 57, 70, 33, 7]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [7, 33, 57, 25, 70]} | {"rule_name": "sum_with_neighbors", "elements": [25, 57, 70, 33, 7], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [95, 20, 8, 82, 60, 45, 49]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [95, 8, 20, 45, 49, 60, 82]} | {"rule_name": "modulo_chain", "elements": [95, 20, 8, 82, 60, 45, 49], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [46, 85, 87, 83, 77, 34]\n\nSorting Rule: Elements ordered by weighted modular arithmetic\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [77, 46, 34, 85, 87, 83]} | {"rule_name": "weighted_modulo", "elements": [46, 85, 87, 83, 77, 34], "rule_description": "Elements ordered by weighted modular arithmetic"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [51, 79, 55, 71, 33, 21, 29, 92]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [21, 92, 29, 33, 51, 55, 71, 79]} | {"rule_name": "lcm_with_constant", "elements": [51, 79, 55, 71, 33, 21, 29, 92], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [80, 25, 98, 54, 21, 92, 61, 90]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [21, 54, 98, 90, 25, 80, 92, 61]} | {"rule_name": "palindrome_distance", "elements": [80, 25, 98, 54, 21, 92, 61, 90], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [6, 4, 70, 77, 25, 83, 56]\n\nSorting Rule: Elements ordered by proximity to Fibonacci sequence\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [4, 6, 56, 25, 83, 77, 70]} | {"rule_name": "fibonacci_proximity", "elements": [6, 4, 70, 77, 25, 83, 56], "rule_description": "Elements ordered by proximity to Fibonacci sequence"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [41, 52, 46, 29, 40, 57, 75, 49, 82]\n\nSorting Rule: Elements ordered by number of divisors\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [29, 41, 49, 46, 57, 82, 52, 75, 40]} | {"rule_name": "divisor_count", "elements": [41, 52, 46, 29, 40, 57, 75, 49, 82], "rule_description": "Elements ordered by number of divisors"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [80, 36, 98, 96, 31, 50]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [36, 80, 96, 50, 31, 98]} | {"rule_name": "lcm_with_constant", "elements": [80, 36, 98, 96, 31, 50], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [89, 1, 44, 11, 36, 63, 68, 77]\n\nSorting Rule: Elements ordered by a hidden circular permutation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [77, 89, 1, 11, 36, 44, 63, 68]} | {"rule_name": "circular_shift", "elements": [89, 1, 44, 11, 36, 63, 68, 77], "rule_description": "Elements ordered by a hidden circular permutation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [5, 65, 75, 49, 26, 82]\n\nSorting Rule: Elements ordered by hidden zone-based rules\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [5, 26, 49, 82, 75, 65]} | {"rule_name": "alternating_zones", "elements": [5, 65, 75, 49, 26, 82], "rule_description": "Elements ordered by hidden zone-based rules"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [86, 32, 39, 7, 63]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [7, 32, 63, 86, 39]} | {"rule_name": "reverse_digits", "elements": [86, 32, 39, 7, 63], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [28, 46, 69, 43, 36, 75, 14, 88, 87, 20]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [36, 46, 28, 14, 20, 43, 87, 75, 69, 88]} | {"rule_name": "sum_with_neighbors", "elements": [28, 46, 69, 43, 36, 75, 14, 88, 87, 20], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [95, 93, 3, 23, 25, 21, 31, 97]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 23, 31, 97, 21, 25, 95, 93]} | {"rule_name": "prime_distance", "elements": [95, 93, 3, 23, 25, 21, 31, 97], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [6, 91, 81, 57, 60, 92, 52, 87, 40, 97]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [6, 87, 52, 81, 91, 97, 40, 57, 60, 92]} | {"rule_name": "bit_operations", "elements": [6, 91, 81, 57, 60, 92, 52, 87, 40, 97], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [55, 46, 81, 14, 22, 11, 92, 2, 39, 85]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [2, 14, 11, 22, 39, 46, 81, 92, 55, 85]} | {"rule_name": "lcm_with_constant", "elements": [55, 46, 81, 14, 22, 11, 92, 2, 39, 85], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [88, 26, 14, 11, 21, 71]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [11, 88, 21, 14, 26, 71]} | {"rule_name": "palindrome_distance", "elements": [88, 26, 14, 11, 21, 71], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [74, 77, 32, 8, 56, 30, 88, 54]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [74, 88, 30, 8, 56, 54, 77, 32]} | {"rule_name": "bit_operations", "elements": [74, 77, 32, 8, 56, 30, 88, 54], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [60, 61, 96, 25, 10, 9, 84, 85, 18]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [84, 10, 18, 85, 96, 9, 60, 25, 61]} | {"rule_name": "bit_operations", "elements": [60, 61, 96, 25, 10, 9, 84, 85, 18], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [49, 4, 1, 87, 32, 8, 70, 71, 15]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [1, 4, 15, 8, 87, 32, 49, 70, 71]} | {"rule_name": "lcm_with_constant", "elements": [49, 4, 1, 87, 32, 8, 70, 71, 15], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [67, 72, 37, 20, 89, 1, 54]\n\nSorting Rule: Elements ordered by proximity to Fibonacci sequence\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [1, 89, 20, 54, 37, 67, 72]} | {"rule_name": "fibonacci_proximity", "elements": [67, 72, 37, 20, 89, 1, 54], "rule_description": "Elements ordered by proximity to Fibonacci sequence"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [48, 5, 99, 86, 42, 2, 97, 78]\n\nSorting Rule: Elements are ordered by a hidden hash function\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [86, 48, 78, 2, 5, 99, 42, 97]} | {"rule_name": "hash_based", "elements": [48, 5, 99, 86, 42, 2, 97, 78], "rule_description": "Elements are ordered by a hidden hash function"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [98, 48, 56, 80, 54]\n\nSorting Rule: Elements ordered by number of divisors\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [98, 54, 56, 48, 80]} | {"rule_name": "divisor_count", "elements": [98, 48, 56, 80, 54], "rule_description": "Elements ordered by number of divisors"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [23, 2, 64, 87, 24, 25, 10, 55]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [2, 23, 10, 24, 25, 55, 87, 64]} | {"rule_name": "prime_distance", "elements": [23, 2, 64, 87, 24, 25, 10, 55], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [71, 82, 67, 21, 54]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [67, 71, 54, 82, 21]} | {"rule_name": "prime_distance", "elements": [71, 82, 67, 21, 54], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [11, 29, 65, 24, 56, 3, 55, 67, 70]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 70, 11, 24, 55, 65, 56, 67, 29]} | {"rule_name": "reverse_digits", "elements": [11, 29, 65, 24, 56, 3, 55, 67, 70], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [14, 92, 82, 42, 37, 7, 91]\n\nSorting Rule: Elements ordered by a hidden circular permutation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [42, 82, 91, 92, 7, 14, 37]} | {"rule_name": "circular_shift", "elements": [14, 92, 82, 42, 37, 7, 91], "rule_description": "Elements ordered by a hidden circular permutation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [42, 4, 47, 75, 76, 43, 90]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [43, 47, 4, 42, 90, 75, 76]} | {"rule_name": "prime_distance", "elements": [42, 4, 47, 75, 76, 43, 90], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [69, 4, 90, 8, 52]\n\nSorting Rule: Elements ordered by weighted modular arithmetic\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [8, 52, 4, 90, 69]} | {"rule_name": "weighted_modulo", "elements": [69, 4, 90, 8, 52], "rule_description": "Elements ordered by weighted modular arithmetic"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [1, 83, 6, 87, 79, 56, 93, 59]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [83, 1, 87, 6, 93, 56, 59, 79]} | {"rule_name": "modulo_chain", "elements": [1, 83, 6, 87, 79, 56, 93, 59], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [77, 68, 79, 58, 72, 66, 28]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [77, 79, 28, 58, 66, 68, 72]} | {"rule_name": "modulo_chain", "elements": [77, 68, 79, 58, 72, 66, 28], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [75, 87, 99, 47, 96, 48, 5, 19]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [5, 47, 75, 48, 87, 96, 19, 99]} | {"rule_name": "digit_operations", "elements": [75, 87, 99, 47, 96, 48, 5, 19], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [30, 80, 53, 21, 18]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [53, 18, 30, 80, 21]} | {"rule_name": "prime_distance", "elements": [30, 80, 53, 21, 18], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [34, 50, 62, 44, 9, 48, 80, 28, 21, 77]\n\nSorting Rule: Elements ordered by a hidden circular permutation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [62, 77, 80, 9, 21, 28, 34, 44, 48, 50]} | {"rule_name": "circular_shift", "elements": [34, 50, 62, 44, 9, 48, 80, 28, 21, 77], "rule_description": "Elements ordered by a hidden circular permutation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [70, 11, 87, 36, 62, 33, 20]\n\nSorting Rule: Elements ordered by weighted modular arithmetic\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [36, 70, 11, 33, 87, 62, 20]} | {"rule_name": "weighted_modulo", "elements": [70, 11, 87, 36, 62, 33, 20], "rule_description": "Elements ordered by weighted modular arithmetic"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [90, 21, 20, 6, 85, 1, 8, 31, 46, 16]\n\nSorting Rule: Elements ordered by proximity to Fibonacci sequence\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [1, 8, 21, 6, 20, 90, 16, 31, 85, 46]} | {"rule_name": "fibonacci_proximity", "elements": [90, 21, 20, 6, 85, 1, 8, 31, 46, 16], "rule_description": "Elements ordered by proximity to Fibonacci sequence"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [85, 9, 94, 66, 55, 80]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [9, 55, 80, 94, 66, 85]} | {"rule_name": "sum_with_neighbors", "elements": [85, 9, 94, 66, 55, 80], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [47, 92, 28, 90, 68, 85, 8, 93, 32, 13]\n\nSorting Rule: Elements ordered by proximity to Fibonacci sequence\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [8, 13, 90, 32, 92, 85, 93, 28, 47, 68]} | {"rule_name": "fibonacci_proximity", "elements": [47, 92, 28, 90, 68, 85, 8, 93, 32, 13], "rule_description": "Elements ordered by proximity to Fibonacci sequence"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [41, 37, 42, 38, 6, 33, 54, 44, 89]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [41, 42, 44, 6, 89, 54, 33, 37, 38]} | {"rule_name": "modulo_chain", "elements": [41, 37, 42, 38, 6, 33, 54, 44, 89], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [70, 98, 76, 35, 69, 36, 5, 42, 7, 18]\n\nSorting Rule: Elements ordered by number of divisors\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [5, 7, 35, 69, 18, 76, 98, 42, 70, 36]} | {"rule_name": "divisor_count", "elements": [70, 98, 76, 35, 69, 36, 5, 42, 7, 18], "rule_description": "Elements ordered by number of divisors"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [64, 83, 73, 44, 22, 77, 49, 67]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [22, 44, 77, 67, 64, 73, 49, 83]} | {"rule_name": "palindrome_distance", "elements": [64, 83, 73, 44, 22, 77, 49, 67], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [57, 10, 75, 25, 89]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [10, 75, 25, 89, 57]} | {"rule_name": "sum_with_neighbors", "elements": [57, 10, 75, 25, 89], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [58, 92, 76, 17, 51, 49, 96, 50, 23]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [58, 17, 76, 23, 92, 96, 49, 50, 51]} | {"rule_name": "modulo_chain", "elements": [58, 92, 76, 17, 51, 49, 96, 50, 23], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [43, 17, 33, 83, 73, 35, 92]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [35, 17, 33, 73, 83, 92, 43]} | {"rule_name": "sum_with_neighbors", "elements": [43, 17, 33, 83, 73, 35, 92], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [53, 17, 66, 16, 84, 20, 65, 50, 90, 45]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [66, 45, 65, 20, 53, 90, 84, 16, 17, 50]} | {"rule_name": "palindrome_distance", "elements": [53, 17, 66, 16, 84, 20, 65, 50, 90, 45], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [71, 3, 39, 12, 50, 93, 80, 66]\n\nSorting Rule: Elements ordered by hidden zone-based rules\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 12, 39, 50, 66, 71, 93, 80]} | {"rule_name": "alternating_zones", "elements": [71, 3, 39, 12, 50, 93, 80, 66], "rule_description": "Elements ordered by hidden zone-based rules"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [14, 8, 74, 31, 20]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [31, 8, 14, 20, 74]} | {"rule_name": "prime_distance", "elements": [14, 8, 74, 31, 20], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [9, 91, 4, 95, 8, 80, 6, 75, 67, 79]\n\nSorting Rule: Elements are ordered by a hidden hash function\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [6, 4, 75, 79, 8, 91, 80, 67, 95, 9]} | {"rule_name": "hash_based", "elements": [9, 91, 4, 95, 8, 80, 6, 75, 67, 79], "rule_description": "Elements are ordered by a hidden hash function"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [41, 20, 79, 44, 32, 39, 77, 91, 40, 51]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [20, 40, 41, 51, 91, 32, 44, 77, 39, 79]} | {"rule_name": "reverse_digits", "elements": [41, 20, 79, 44, 32, 39, 77, 91, 40, 51], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [92, 23, 5, 69, 79, 51, 13]\n\nSorting Rule: Elements ordered by number of divisors\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [5, 13, 23, 79, 51, 69, 92]} | {"rule_name": "divisor_count", "elements": [92, 23, 5, 69, 79, 51, 13], "rule_description": "Elements ordered by number of divisors"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [52, 30, 58, 98, 82, 10, 1, 91]\n\nSorting Rule: Elements ordered by weighted modular arithmetic\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [1, 91, 58, 82, 30, 52, 10, 98]} | {"rule_name": "weighted_modulo", "elements": [52, 30, 58, 98, 82, 10, 1, 91], "rule_description": "Elements ordered by weighted modular arithmetic"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [97, 77, 76, 44, 33, 58, 23]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [97, 23, 33, 44, 58, 76, 77]} | {"rule_name": "modulo_chain", "elements": [97, 77, 76, 44, 33, 58, 23], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [69, 82, 34, 41, 57]\n\nSorting Rule: Elements are ordered by a hidden hash function\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [41, 34, 82, 57, 69]} | {"rule_name": "hash_based", "elements": [69, 82, 34, 41, 57], "rule_description": "Elements are ordered by a hidden hash function"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [73, 29, 68, 62, 51, 15, 45]\n\nSorting Rule: Elements are ordered by a hidden hash function\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [62, 15, 73, 68, 29, 45, 51]} | {"rule_name": "hash_based", "elements": [73, 29, 68, 62, 51, 15, 45], "rule_description": "Elements are ordered by a hidden hash function"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [30, 65, 80, 34, 75]\n\nSorting Rule: Elements ordered by hidden zone-based rules\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [30, 34, 65, 80, 75]} | {"rule_name": "alternating_zones", "elements": [30, 65, 80, 34, 75], "rule_description": "Elements ordered by hidden zone-based rules"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [57, 24, 67, 88, 4, 86, 94]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [57, 4, 24, 67, 88, 94, 86]} | {"rule_name": "bit_operations", "elements": [57, 24, 67, 88, 4, 86, 94], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [52, 77, 71, 75, 5, 31, 81, 62]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [5, 77, 31, 75, 52, 62, 81, 71]} | {"rule_name": "palindrome_distance", "elements": [52, 77, 71, 75, 5, 31, 81, 62], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [70, 24, 62, 94, 36, 39, 74, 21, 61, 69]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [21, 70, 24, 36, 39, 61, 62, 69, 74, 94]} | {"rule_name": "lcm_with_constant", "elements": [70, 24, 62, 94, 36, 39, 74, 21, 61, 69], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [70, 60, 76, 71, 68, 54, 56, 5]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [54, 5, 56, 60, 71, 76, 70, 68]} | {"rule_name": "digit_operations", "elements": [70, 60, 76, 71, 68, 54, 56, 5], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [88, 14, 79, 50, 44, 99, 72]\n\nSorting Rule: Elements ordered by a hidden circular permutation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [44, 50, 72, 79, 88, 99, 14]} | {"rule_name": "circular_shift", "elements": [88, 14, 79, 50, 44, 99, 72], "rule_description": "Elements ordered by a hidden circular permutation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [48, 34, 22, 81, 57, 29]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [48, 22, 34, 29, 57, 81]} | {"rule_name": "lcm_with_constant", "elements": [48, 34, 22, 81, 57, 29], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [64, 40, 9, 81, 5]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [9, 40, 5, 64, 81]} | {"rule_name": "sum_with_neighbors", "elements": [64, 40, 9, 81, 5], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [84, 53, 88, 13, 5, 3, 18, 30, 73]\n\nSorting Rule: Elements ordered by weighted modular arithmetic\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [88, 3, 13, 84, 5, 73, 30, 18, 53]} | {"rule_name": "weighted_modulo", "elements": [84, 53, 88, 13, 5, 3, 18, 30, 73], "rule_description": "Elements ordered by weighted modular arithmetic"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [67, 80, 7, 26, 9, 42, 65, 17]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [26, 42, 9, 7, 67, 17, 65, 80]} | {"rule_name": "bit_operations", "elements": [67, 80, 7, 26, 9, 42, 65, 17], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [35, 41, 70, 59, 33, 14, 57, 37, 15]\n\nSorting Rule: Elements are ordered by a hidden hash function\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [15, 57, 35, 41, 33, 37, 59, 14, 70]} | {"rule_name": "hash_based", "elements": [35, 41, 70, 59, 33, 14, 57, 37, 15], "rule_description": "Elements are ordered by a hidden hash function"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [97, 91, 36, 95, 11, 86]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [11, 36, 86, 95, 97, 91]} | {"rule_name": "digit_operations", "elements": [97, 91, 36, 95, 11, 86], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [20, 94, 22, 13, 68, 18]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [13, 18, 20, 22, 94, 68]} | {"rule_name": "bit_operations", "elements": [20, 94, 22, 13, 68, 18], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [35, 97, 10, 22, 81, 67, 53, 33, 55, 4]\n\nSorting Rule: Elements ordered by number of divisors\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [53, 67, 97, 4, 10, 22, 33, 35, 55, 81]} | {"rule_name": "divisor_count", "elements": [35, 97, 10, 22, 81, 67, 53, 33, 55, 4], "rule_description": "Elements ordered by number of divisors"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [25, 36, 64, 40, 89, 34, 49, 16, 80, 72]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [16, 25, 34, 36, 40, 49, 64, 72, 80, 89]} | {"rule_name": "lcm_with_constant", "elements": [25, 36, 64, 40, 89, 34, 49, 16, 80, 72], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [24, 41, 65, 68, 43, 64, 59]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [24, 41, 43, 64, 65, 68, 59]} | {"rule_name": "digit_operations", "elements": [24, 41, 65, 68, 43, 64, 59], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [96, 36, 57, 87, 84, 73, 17, 66]\n\nSorting Rule: Elements ordered by proximity to Fibonacci sequence\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [36, 57, 87, 17, 84, 96, 66, 73]} | {"rule_name": "fibonacci_proximity", "elements": [96, 36, 57, 87, 84, 73, 17, 66], "rule_description": "Elements ordered by proximity to Fibonacci sequence"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [7, 82, 75, 79, 41, 63, 97]\n\nSorting Rule: Elements ordered by cascading modulo operations (hidden moduli)\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [7, 63, 79, 82, 75, 97, 41]} | {"rule_name": "modulo_chain", "elements": [7, 82, 75, 79, 41, 63, 97], "rule_description": "Elements ordered by cascading modulo operations (hidden moduli)"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [71, 79, 19, 97, 40, 39, 99]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [40, 71, 97, 19, 39, 79, 99]} | {"rule_name": "reverse_digits", "elements": [71, 79, 19, 97, 40, 39, 99], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [35, 49, 96, 72, 93, 24]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [24, 72, 96, 35, 49, 93]} | {"rule_name": "lcm_with_constant", "elements": [35, 49, 96, 72, 93, 24], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [45, 42, 69, 78, 2, 97]\n\nSorting Rule: Elements ordered by a hidden bitwise operation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [69, 97, 45, 78, 2, 42]} | {"rule_name": "bit_operations", "elements": [45, 42, 69, 78, 2, 97], "rule_description": "Elements ordered by a hidden bitwise operation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [82, 68, 43, 81, 62, 75]\n\nSorting Rule: Elements ordered by number of divisors\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [43, 62, 82, 81, 68, 75]} | {"rule_name": "divisor_count", "elements": [82, 68, 43, 81, 62, 75], "rule_description": "Elements ordered by number of divisors"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [17, 96, 30, 46, 33, 27]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [30, 33, 46, 27, 17, 96]} | {"rule_name": "digit_operations", "elements": [17, 96, 30, 46, 33, 27], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [39, 87, 72, 62, 85, 52, 7, 33, 81, 38]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [7, 33, 87, 52, 85, 62, 81, 38, 39, 72]} | {"rule_name": "palindrome_distance", "elements": [39, 87, 72, 62, 85, 52, 7, 33, 81, 38], "rule_description": "Elements ordered by distance to palindromic numbers"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [4, 22, 67, 41, 99, 16, 42]\n\nSorting Rule: Elements ordered by weighted modular arithmetic\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [99, 22, 67, 16, 42, 4, 41]} | {"rule_name": "weighted_modulo", "elements": [4, 22, 67, 41, 99, 16, 42], "rule_description": "Elements ordered by weighted modular arithmetic"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [66, 27, 64, 25, 92, 72, 48, 53, 22, 30]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [53, 22, 30, 48, 66, 72, 25, 27, 64, 92]} | {"rule_name": "prime_distance", "elements": [66, 27, 64, 25, 92, 72, 48, 53, 22, 30], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [3, 74, 69, 45, 24, 63]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 63, 24, 74, 45, 69]} | {"rule_name": "reverse_digits", "elements": [3, 74, 69, 45, 24, 63], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [15, 91, 40, 74, 37, 51, 5]\n\nSorting Rule: Elements ordered by hidden zone-based rules\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [5, 15, 40, 37, 51, 91, 74]} | {"rule_name": "alternating_zones", "elements": [15, 91, 40, 74, 37, 51, 5], "rule_description": "Elements ordered by hidden zone-based rules"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [62, 54, 41, 39, 87, 29, 96]\n\nSorting Rule: Elements ordered by a hidden digit-based calculation\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [41, 54, 62, 87, 39, 96, 29]} | {"rule_name": "digit_operations", "elements": [62, 54, 41, 39, 87, 29, 96], "rule_description": "Elements ordered by a hidden digit-based calculation"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [39, 80, 55, 67, 3, 58, 44, 53]\n\nSorting Rule: Elements ordered by their digit-reversed values\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 80, 53, 44, 55, 67, 58, 39]} | {"rule_name": "reverse_digits", "elements": [39, 80, 55, 67, 3, 58, 44, 53], "rule_description": "Elements ordered by their digit-reversed values"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [94, 49, 45, 61, 82, 9, 64, 93, 34, 13]\n\nSorting Rule: Elements ordered by hidden zone-based rules\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [9, 13, 49, 45, 34, 61, 64, 94, 93, 82]} | {"rule_name": "alternating_zones", "elements": [94, 49, 45, 61, 82, 9, 64, 93, 34, 13], "rule_description": "Elements ordered by hidden zone-based rules"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [66, 45, 7, 29, 58, 54, 14]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [66, 7, 14, 29, 45, 54, 58]} | {"rule_name": "lcm_with_constant", "elements": [66, 45, 7, 29, 58, 54, 14], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [13, 11, 16, 80, 28, 61, 99, 51, 89, 58]\n\nSorting Rule: Elements ordered by LCM with a hidden constant\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [16, 28, 80, 11, 13, 58, 51, 61, 89, 99]} | {"rule_name": "lcm_with_constant", "elements": [13, 11, 16, 80, 28, 61, 99, 51, 89, 58], "rule_description": "Elements ordered by LCM with a hidden constant"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [42, 37, 71, 12, 41, 62]\n\nSorting Rule: Elements ordered by distance to nearest prime number\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [37, 41, 71, 12, 42, 62]} | {"rule_name": "prime_distance", "elements": [42, 37, 71, 12, 41, 62], "rule_description": "Elements ordered by distance to nearest prime number"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [4, 1, 17, 13, 46, 26, 75, 82, 35]\n\nSorting Rule: Elements ordered by hidden pairwise relationships\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [1, 13, 46, 26, 4, 75, 82, 17, 35]} | {"rule_name": "sum_with_neighbors", "elements": [4, 1, 17, 13, 46, 26, 75, 82, 35], "rule_description": "Elements ordered by hidden pairwise relationships"} |
hash_values | [{"role": "system", "content": "You are a helpful assistant that solves sorting problems. You will be given a list of numbers and a sorting rule. Output the sorted array as a JSON list, e.g. [3, 1, 4, 1, 5]. Do not include any other text after the JSON list."}, {"role": "user", "content": "You need to sort the following array:\n\nArray: [40, 3, 95, 12, 64, 56, 32, 45, 71]\n\nSorting Rule: Elements ordered by distance to palindromic numbers\n\nYou can ONLY use the compare_elements tool to determine the order of elements. You need to:\n1. Design a sorting algorithm\n2. Call compare_elements to compare elements\n3. Return the final sorted array\n\nTool Definition:\n{\n \"name\": \"compare_elements\",\n \"description\": \"Compare two elements to determine their relative order. This is the ONLY tool you can use to determine element ordering.\",\n \"parameters\": {\n \"type\": \"object\",\n \"properties\": {\n \"element_a\": {\n \"type\": \"number\",\n \"description\": \"First element to compare\"\n },\n \"element_b\": {\n \"type\": \"number\",\n \"description\": \"Second element to compare\"\n }\n },\n \"required\": [\n \"element_a\",\n \"element_b\"\n ]\n }\n}\n\nEach tool call should be in JSON format:\n{\"tool\": \"compare_elements\", \"arguments\": {\"element_a\": X, \"element_b\": Y}}\n\nBegin sorting now.\n"}] | opaque-tool-sorting | {"style": "rule", "ground_truth": [3, 12, 32, 45, 56, 64, 40, 95, 71]} | {"rule_name": "palindrome_distance", "elements": [40, 3, 95, 12, 64, 56, 32, 45, 71], "rule_description": "Elements ordered by distance to palindromic numbers"} |
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