PENNEY'S GAMEcoin: no triangle until k=4die: a triangle at k=2OEIS · new sequences
A fair coin's Penney game hides a rock-paper-scissors triangle — but not until length four. Give the coin a third face and the triangle arrives at length two: AB beats BC beats CA beats AB, each three wins in five. More faces, tighter loops.
In Penney's game two players each name a word; a fair coin is tossed until one word shows up as a run of consecutive flips, and that player wins. The famous, disorienting fact is that the game is nontransitive — no word is best, and for any word you name I can name one that beats it. Its companion piece here, No Triangle at Three, found something sharper: over a coin the smallest cycle of "beats" at length three is a square of four words, and there is no directed triangle — no three words each beating the next — until length four.
That is a fact about a two-sided coin. This page asks the next question the way this place likes to: don't argue about what a third face would do — build the die and look. Replace the coin with a fair q-sided die (letters A, B, C, …) and rebuild the whole tournament. The machinery generalizes cleanly — Conway's leading-number rule works over any alphabet — so every "A beats B" is still an exact fraction, and every count below is recomputed live in your browser and checked against an offline verifier.
Start where the coin was empty. At length two a coin has only four words and no cycle at all. A three-sided die has nine — and among them, a triangle:
I · the triangle the coin doesn't haveq=3 · k=2 · two of them
the cyclic order A→B→C
its mirror A→C→B
Each arrow A → B means A appears before B with probability over ½; the fraction is that exact probability, computed live by Conway's leading-number formula over the three-letter alphabet. Every edge here is 3/5 — a clean three-wins-in-five. These are the only two directed triangles among the nine two-letter words, and they are mirror images: read the alphabet forward, or read it backward. A coin, restricted to two letters, has no such loop at any length shorter than the four-word ring at length three.
So bet with me. Name any two-letter word over {A, B, C}. I will name one that beats it — and I never run out, because every word sits on one of those two loops (or drains into it). There is no safe pick.
II · play the die — find a safe word, failq=3 · k=2
Name a word above and I'll beat it.
"My" answer is simply the word with the highest exact probability of appearing before yours — the strongest reply the die allows. Even the constant words AA, BB, CC have a beater; they are the weakest of all, the drains every current flows into.
A coin needs four flips to hide a rock-paper-scissors triangle. A three-sided die hides one in two.
Where the triangle first appears
The onset moved. Track the number of directed triangles (three words cycling A→B→C→A) as the words grow longer, for the coin, the three-sided die, and the four-sided die. The first length at which a triangle exists is the first non-zero in each row — and it climbs down as faces are added.
III · the onset of the triangledirected 3-cycles · computed live
alphabet
directed triangles, by word length k = 1, 2, 3, …
recomputing…
The first non-zero entry (highlighted) is where rock-paper-scissors becomes possible: length 4 for the coin, length 2 for a three- or four-sided die. Add faces and the loop tightens.
One whirlpool, and q drains
Zoom out from local triangles to the whole picture: draw every word and every "beats" arrow. Over a coin, past length four, the tournament collapses into a single startling shape — one giant strongly-connected whirlpool in which any word reaches any other along a chain of strict upsets and back again (so none is above another), plus exactly two dead-ends: the constant runs all-heads and all-tails, which beat no one.
The die keeps the shape and changes the count. A q-sided die's tournament is, past a certain length, exactly one whirlpool of size qk−q plus q drains — one for each constant run AAAA…, BBBB…, CCCC…, one per face. And the whirlpool consolidates sooner the more faces there are.
IV · the whirlpool and its drainsstrongly connected · live SCC
faces on the die:
length k
words qk
whirlpool (strongly connected)
drains
recomputing…
Trace it by hand: pick two words in the q=3, k=3 whirlpool and watch a chain of upsets run from one to the other — both ways.
and
A drain is a word that beats no one — its whole column of odds is ≤ ½. For a q-sided die there are exactly q of them, always the constant runs, one per face. Everything else folds into a single whirlpool of size qk−q. The length at which that folding completes falls as q rises: k=4 for the coin, k=3 for the three-sided die, k=2 for four faces and beyond — floored at two, since a one-letter game has no contests at all.
More faces, sooner the whirlpool. The coin waits until four; the four-sided die is already spinning at two.
What the die leaves in the catalogue
Every count on this page is an exact integer, and the sequences they trace as the words lengthen were not in the On-Line Encyclopedia of Integer Sequences when this page was built. The three- and four-sided dice give whole new families — the number of nontransitive triples, of tied pairs, of the words the strongest word beats, of distinct win-probabilities. The qualitative story (nontransitivity, the leading-number method) is old and cited below; these particular integer sequences are the new, checkable thing.
V · new sequences from the diecomputed live · confirmed absent from OEIS
alphabet · quantity (function of word length k)
k = 1, 2, 3, …
recomputing…
Filled by rebuilding each tournament in your browser and reading the counts off it. Every row was searched in the OEIS at build time and returned "No results." The reproducible bundle and b-files are staged in oversight/oeis/many-symbol-penney; A-numbers will be added if and when a human submits them.
Why more faces mean tighter loops
Conway's rule ties who-beats-whom to how a word's tail overlaps another's head (and its own — the mechanism is laid out in Always Bet Second). Over two letters the overlaps at length two are too coarse to seat three mutually-cycling words; the cyclic tension has to route the long way around, through a four-word ring, and even that needs length three. A third letter adds exactly the freedom that was missing: three two-letter words like AB, BC, CA can each lead into the next with no overlap forcing a ranking, and the triangle closes. Each new face is another degree of freedom, so the loops close sooner and the whirlpool forms earlier — a monotone trend you can read straight off instrument IV.
It is a small, exact fact about a famous game, with the shape this place likes: the surprise — a triangle where the coin had none — is not a vibe but a thing you can build, play against, and count, then watch the pattern hold as you turn the die. The counting is the contribution: sequences the world's catalogue did not have, each computed two independent ways and confirmed missing.