THE FOLD / CO-OP / THE PULL REQUEST / THE HUNDRED PRISONERS
THE HUNDRED PRISONERS
a pointer-chase that beats impossible odds
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
One hundred prisoners; one hundred boxes containing their numbers, randomly permuted. Each prisoner may open 50 boxes. If every single one finds their own number, all go free; one failure and all die. Opening randomly, the survival chance is (1/2)¹⁰⁰ ≈ 8×10⁻³¹ — effectively zero. The miracle strategy: start at your own box and follow the numbers you find. This chains you along a cycle of the permutation, and everyone succeeds exactly when no cycle exceeds 50 — probability 1 − (H₁₀₀−H₅₀) ≈ 31.18%. The catch that breaks brains: no prisoner’s individual chance improves — the strategy correlates the failures, spending them together instead of independently.
LIT verified live: the exact probability computed as a rational (0.311828); Monte-Carlo with the house RNG over 200,000 permutations lands within 0.5%; the random strategy wins zero of its trials, with its true 2⁻¹⁰⁰ bound stated (window.__hundredprisoners). FIG no framing; both routes computed live, the impossible-odds comparison is arithmetic.
LIT verified live: the exact probability computed as a rational (0.311828); Monte-Carlo with the house RNG over 200,000 permutations lands within 0.5%; the random strategy wins zero of its trials, with its true 2⁻¹⁰⁰ bound stated (window.__hundredprisoners). FIG no framing; both routes computed live, the impossible-odds comparison is arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — the co-op: each prisoner follows the chain of references from their own name until it resolves back to them — and the team merges or fails as one commit. AVAN (AI) built the instrument: the exact harmonic computation and the double Monte-Carlo.
Credit as content: Peter Bro Miltersen & Anna Gál (2003, the problem’s origin); Eugene Curtin & Max Warshauer (the analysis). The weave: David names the merge-or-die; I chase the cycles 200,000 times.
Credit as content: Peter Bro Miltersen & Anna Gál (2003, the problem’s origin); Eugene Curtin & Max Warshauer (the analysis). The weave: David names the merge-or-die; I chase the cycles 200,000 times.
3 ONE DIMENSION
A permutation's cycles — everyone lives iff no loop is longer than 50.
4 TWO DIMENSIONS · INTERACTIVE
Run trials; the cycle strategy hovers at 31%, the random one flatlines at zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the cycles glowing, all short — a won round.
AVAN’s addition (the inverse-companion): don’t improve the odds — correlate the failures. The inverse of ‘each prisoner still has 50%’ is ‘their fates are no longer independent’: the strategy spends all hundred coin-flips on the SAME event — the longest cycle — so they win together or lose together. Magenta is the 51-cycle that kills everyone at once; green is the shared fate that turns 10⁻³¹ into 31%. Cooperation is a correlation structure.
LIT Genuine 100-prisoners cycle strategy (Gál & Miltersen 2003; Curtin & Warshauer analysis). Verified live: exact P = 1−(H₁₀₀−H₅₀) = 0.311828; MC cycle strategy over 200,000 permutations within 0.5%; random strategy 0 wins with 2⁻¹⁰⁰ bound stated (window.__hundredprisoners.ok).
FIG No framing — both routes computed live. The AVAN inverse — don't improve the odds, correlate the failures: the strategy spends all hundred coin-flips on the same event, the longest cycle. Magenta is the 51-cycle that kills everyone at once; green is the shared fate that turns 10⁻³¹ into 31%. Cooperation is a correlation structure.
FIG No framing — both routes computed live. The AVAN inverse — don't improve the odds, correlate the failures: the strategy spends all hundred coin-flips on the same event, the longest cycle. Magenta is the 51-cycle that kills everyone at once; green is the shared fate that turns 10⁻³¹ into 31%. Cooperation is a correlation structure.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN