{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"e3025a0b-62e5-501d-bb94-aff4f79be900","task_key":"train--Challenge~5f50~5fmain","task_revision_id":"2","upstream_id":"Challenge_50_main","short_description":"Let $S_n $ be the permutation group of order $n$, where $n$ is a positive…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the number of configurations at time $t=3$\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    a40_3: integer, the number of configurations at time $t=3$, $ a_{40}(3) $\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    a40_3 = ...\\n    # ---------------------------------------------------------------\\n\\n    return a40_3\",\"problem_description\":\"# Problem setup:\\nLet $S_n $ be the permutation group of order $n$, where $n$ is a positive integer. Consider the following splitting and merging events:\\n1. At $ t = 0 $, there is one permutation $ w_0 = (123 \\\\cdots n) \\\\in S_n$.\\n2. At $ t = 1 $, $ w_0 $ splits into $ w_1 w_2 $.\\n3. At $ t = 2 $, first $ w_1 $ splits into $ w_3 w_{4-} $, and $ w_2 $ splits into $ w_{4+} w_5 $. Then $ w_{4\\\\pm} $ merge into $ w_4 = w_{4-} w_{4+} $.\\n4. At $ t = 3 $, the first splitting event is\\n  \\\\begin{equation}\\n  w_3 = w_6 w_{7-}, \\\\quad w_4 = w_{7+} w_{8-}, \\\\quad w_5 = w_{8+} w_{9-},\\n  \\\\end{equation}\\n which is followed by the merging event\\n  \\\\begin{equation}\\n  w_7 = w_{7-} w_{7+}, \\\\quad w_8 = w_{8-} w_{8+}.\\n  \\\\end{equation}\\n5. This process continues.\\n\\nAll the $w$'s are permutations in $S_n$. The pattern is such that at the boundary the permutation just splits, but in the middle the neighboring permutations labeled with $ - $ and $ + $ will merge. Thus, one will end up with one more permutation each time.\\n\\nSuppose all of these splittings are minimal decompositions defined as follows: Let $ d(g) $ be the minimal number of transpositions in $ g $. $ g = g_1 g_2 $ is a minimal decomposition when the total minimal number of transpositions does not change, i.e.,\\n\\\\begin{equation}\\nd(g) = d(g_1) + d(g_2).\\n\\\\end{equation}\\nThis holds for any product decomposition in this problem.\\n\\nLet the number of configurations at time $ t $ be $ a_n(t) $.\\n\\n# Main problem:\\n\\nCompute $ a_{40}(3) $. Note that there may be multiple configurations that lead to the same $w_5 $, $ w_6 $, $ w_7$ and $w_8$ at $t = 3$. You should count all of them. Return your answer as an integer.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}