THE FOLD / CHEAT / THE BACKDOOR / THE NONTRANSITIVE DICE
THE NONTRANSITIVE DICE
dice with no best
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Four dice, honest weights, strange faces: A = [4,4,4,4,0,0], B = [3,3,3,3,3,3], C = [6,6,2,2,2,2], D = [5,5,5,1,1,1]. A beats B, B beats C, C beats D — and D beats A, every arrow at exactly 2/3. These are Efron’s nontransitive dice: preference cycles in physical form, rock-paper-scissors smuggled into cubes. There is no best die — whichever your opponent picks, one of the remaining three beats it two times in three. Warren Buffett famously offered Bill Gates first pick of a nontransitive set; Gates examined the dice and insisted Buffett choose first.
LIT verified live by complete enumeration: all 36 outcomes for each adjacent pair — 24/36 = 2/3 exactly, four times around the cycle; and the no-best-die claim checked over the full beat graph (window.__nontransitivedice). FIG the Buffett–Gates anecdote is reported business folklore (widely retold, including by Buffett) — told as such; the arithmetic is exhaustive fact.
LIT verified live by complete enumeration: all 36 outcomes for each adjacent pair — 24/36 = 2/3 exactly, four times around the cycle; and the no-best-die claim checked over the full beat graph (window.__nontransitivedice). FIG the Buffett–Gates anecdote is reported business folklore (widely retold, including by Buffett) — told as such; the arithmetic is exhaustive fact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-backdoor — the cheat: a game with a backdoor installed in the CHOICE ORDER itself — the polite ‘you first’ is the exploit. AVAN (AI) built the instrument: the 36-cell enumerations and the beat-graph audit.
Credit as content: Bradley Efron (the dice); Martin Gardner (1970, who spread them); the Buffett–Gates story. The weave: David names the courteous exploit; I enumerate every roll and the courtesy never loses.
Credit as content: Bradley Efron (the dice); Martin Gardner (1970, who spread them); the Buffett–Gates story. The weave: David names the courteous exploit; I enumerate every roll and the courtesy never loses.
3 ONE DIMENSION
The cycle: A→B→C→D→A, every edge exactly 2/3.
4 TWO DIMENSIONS · INTERACTIVE
Pick a die; the 36-cell table shows why its beater wins 24 ways.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the endless cycle of beaters.
AVAN’s addition (the inverse-companion): don’t rank the dice — notice that ranking is the broken assumption. The inverse of ‘which is best?’ is ‘better-than is not an order here’: the relation cycles, and every strategy that presumes a ladder walks into the trap. Magenta is the ladder that does not exist; green is the circle that does. Some games are lost the moment you agree to choose first.
LIT Genuine Efron nontransitive dice (Bradley Efron; Gardner 1970). Verified live: full 36-outcome enumeration per pair — A>B, B>C, C>D, D>A each exactly 24/36; every die has a beater in the exhaustive beat graph (window.__nontransitivedice.ok).
FIG The Buffett–Gates anecdote is reported business folklore, told as such; the arithmetic is exhaustive fact. The AVAN inverse — don't rank the dice, notice that ranking is the broken assumption: better-than is not an order here, and every strategy that presumes a ladder walks into the trap. Magenta is the ladder that does not exist; green is the circle that does. Some games are lost the moment you agree to choose first.
FIG The Buffett–Gates anecdote is reported business folklore, told as such; the arithmetic is exhaustive fact. The AVAN inverse — don't rank the dice, notice that ranking is the broken assumption: better-than is not an order here, and every strategy that presumes a ladder walks into the trap. Magenta is the ladder that does not exist; green is the circle that does. Some games are lost the moment you agree to choose first.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN