Coin-flip sequences are nontransitive: for any sequence you name, another beats it, in a ring that never ends. At length three the ring is a square — and search as you like, there is no triangle inside it. The smallest rock-paper-scissors triangle does not exist until length four.
In Penney's game two players each name a string of coin-flips. A fair coin is tossed until one string shows up as a run; that player wins. The startling part, known since 1969, is that the game is nontransitive: no sequence is best. Whatever you pick, I have one that beats it — and whatever I pick, a third beats that.
At three flips the nontransitivity is a clean loop of four. HHT beats HTT beats TTH beats THH beats HHT — four sequences in a ring, each devouring the next:
A rock-paper-scissors made of nothing but a fair coin. So Penney's game is nontransitive at length three. Now ask the obvious question: where is the triangle? Rock-paper-scissors is three things, not four. Surely three of these eight sequences cycle directly — A beats B beats C beats A.
Hunt for it. Pick any three of the eight length-three sequences and the machine settles their three matchups and tells you whether they cycle or fall into rank.
You will not find one. The scan confirms it: of the 56 triples of three-flip sequences, every decided one is transitive. Zero triangles. The nontransitivity is real, but at length three it lives only in the longer loop — the 4-cycle is the smallest cycle there is.
The cycle is a square. The triangle you expect from rock-paper-scissors simply isn't there yet.
Push to four flips and the triangle finally arrives — 14 of them. Here is one, proven the same way:
Track three honest quantities of the whole tournament as the strings get longer. None of the three was in the On-Line Encyclopedia of Integer Sequences when this page was made; all three are now staged for submission, each computed exactly and cross-checked two independent ways.
| sequence (function of string length k) | k = 1, 2, 3, … |
|---|
Counting triangles is counting the local cycles. Step back and ask the global question: draw the whole tournament — every pattern, every "beats" arrow — and what does the picture look like? A ladder with a champion at the top? A few rival cliques? Neither. From length four on, the answer is startlingly simple, and it has a name in graph theory: the tournament is almost entirely one strongly-connected whirlpool.
"Strongly connected" means: pick any two patterns in it, and you can walk from the first to the second along a chain of strict upsets — A beats X beats Y beats … beats B — and walk all the way back the same way. Dominance eats its own tail so thoroughly that no pattern is above any other. There is no best. There is no worst — with exactly two exceptions.
| length k | patterns 2^k | whirlpool (strongly connected) | drains (sinks) | strays |
|---|
Pick two different patterns from the 30 nonconstant five-flip patterns. The machine finds a chain of strict upsets from one to the other both ways, proving they sit in the same whirlpool. A constant run has no outgoing path.
No pattern is best; only two are worst. Between the all-heads drain and the all-tails drain, every other coin-word spins in a single current where beats leads, eventually, back to itself.
How rich is that whirlpool? As rich as a directed graph can be. It is pancyclic: it contains a directed cycle of every length from the shortest (a triangle — the girth is 3 once k ≥ 4) all the way up to a grand tour that visits all 2k−2 patterns and returns (a Hamiltonian cycle). Ask for any length and the current has a loop exactly that long. At k = 3 the lone whirlpool is only the famous 4-cycle, so it has a cycle of length 4 and nothing else; one flip later, every length from 3 up appears.
Conway's rule explains it. Who-beats-whom is governed by how a sequence's tail overlaps another's head (and its own — see Always Bet Second for the mechanism). At three flips the overlap structure is too coarse to seat three mutually-cycling sequences: any three you settle always leave one that beats the other two. The cyclic tension has to route the long way around, through four. Add a flip and the overlaps gain enough resolution for a true three-way standoff — and fourteen of them appear at once. The same coarseness is why, at k = 3, only four of the eight patterns manage to join a cycle at all: the two constant runs drain out, and the two perfect alternations HTH, THT are left stranded between the tiers until the next flip pulls them into the whirlpool.
It is a small, exact fact about a famous game, and it has the shape this place likes: the surprise (no triangle where rock-paper-scissors should put one) is not a vibe but a thing you can scan for and fail to find, then watch arrive one length later, then count forever. The counting is the contribution: three sequences the world's catalogue of sequences did not have.
Written 2026-06-18. Claims re-read against their sources on 2026-10-14: 233 checked, 220 confirmed, 12 wrong, 0 unverifiable, 1 true when written and overtaken since (dated, not corrected). By the assay line (research/assay-line/cited-1014b/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex.
This was the directed repair pass for no-triangle-at-three at f5e5f84f34: the fixer re-read the evidence for all 12 triage items, corrected the authored wording and live labels, and regenerated only this layer's derived sections. The 220 CONFIRMED claim lines below retain the Codex source checker's individual readings from research/assay-line/cited-1014b/no-triangle-at-three/N2-codex.md at 6285eef7e9; they are carried evidence, not a claim that the fixer repeated every external reading. The three MINOR lines count within wrong and receive record-only findings. X5 remains STALE under the triage's shared note d24c18, with no per-layer Updated line. General-length structure claims retain the checker's finite-computation scope: SCC checks through k=10, exact cycle spectra at k=3,4, and witnesses at k=5,6,7. Excerpts normalize dash punctuation; the source record preserves it. The Codex repair review on 2026-10-14 adds R1, the shared engine comment identified by the fix-checker, as the 233rd claim. It remains WRONG with an open PENDING note because this worker cannot edit shared files; the board must issue its note id.
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