Three Dice, Each Beaten by the Next

Three six-sided dice are laid out as circles, each with its six faces printed beside it. Two dice are compared by throwing both and seeing which shows the higher face. Rather than throwing them, the laboratory counts: it takes every face of one die against every face of the other, 36 pairings in all, and records who wins each.

An arrow runs from the die that wins more of its pairings to the die it beats, and carries the two counts. On the opening dice the arrows run A to B, B to C and C to A, round the triangle. No die is at the top. Whichever one you pick, another die beats it.

The tempting explanation is that beating means being bigger, so a chain of beating must climb. It cannot climb round in a circle. That explanation is fine for numbers. It is wrong for dice, because a die is not one number but six, and two dice are compared by laying their six faces against each other.

Look at the totals view. The three dice have the same average face, 5, so none of them is bigger on average. What differs is the shape. A has two very high faces, two middling and two low. B has two very low faces and four high ones. C is spread evenly through the middle. When A meets B, the question is which faces of A line up above which faces of B, and when B meets C it is a different set of pairings entirely. Nothing makes the first set of pairings predict the second.

A ranking needs one number for each thing ranked. Put the things in order by their number, and anything above is above everything below. A comparison that only ever looks at two things at a time, and decides by counting, does not hand out numbers. It hands out a verdict for each pair, and nothing forces three such verdicts to agree with an order.

The dice make this concrete. The win rate of A against B is a fact about the pair, not about A. Change B and A's rate against it changes, and a die that looks strong against one opponent can look weak against the next. The loop is what that independence looks like when it is drawn as arrows.

The second throw makes the same point from another side. Adding two throws turns each die into a new die with 36 equally likely faces, and the new dice are compared the same way. Which arrow points where was never a property of the original six faces alone, and here it reorders: A stops beating B.

For two dice $X$ and $Y$ with faces $x_1,\dots ,x_6$ and $y_1,\dots ,y_6$, count the pairings $(i,j)$ in which $X$ wins and in which it loses:

Pairings with $x_i=y_j$ are ties and count for neither side. Say that $X$ beats $Y$ when $W(X,Y)>L(X,Y)$, which with six faces each means winning more than half of the decided pairings. With two throws each, replace the faces by the 36 possible sums of two faces of the same die, so there are $36\times 36=1296$ pairings.

Three dice $A$, $B$, $C$ are intransitive when $A$ beats $B$, $B$ beats $C$ and $C$ beats $A$, or when all three results run the other way round. Then no die beats both of the others, so no die can be placed first in any order that respects the results.

To show that "beats" need not be transitive it is enough to give one set of dice and count. Here is the count for the opening dice with one throw each. The plate above has A against B: A wins $4+4+12=20$ pairings and loses $8+8+0=16$.

For B against C, the two 1s of B lose to every face of C, which is 12 losses. The two 6s of B beat C's 3, 3, 5, 5 and lose to its 7, 7, giving 8 wins and 4 losses. The two 8s beat all of C, 12 wins. Total: $0+8+12=20$ wins and $12+4+0=16$ losses.

For C against A, the two 3s of C beat A's 2, 2 and lose to 4, 4, 9, 9, giving 4 wins and 8 losses. The two 5s beat 2, 2, 4, 4 and lose to 9, 9, giving 8 wins and 4 losses. The two 7s do the same, 8 wins and 4 losses. Total: $4+8+8=20$ wins and $8+4+4=16$ losses.

So A beats B, B beats C and C beats A, each 20 to 16, and no die is best. That is a proof of existence, and it uses only these 108 pairings.

With two throws each the space is larger, 1,296 pairings for each pair of dice, and the page does not argue it by hand. It counts all of them exactly: A wins 592 and loses 704 against B, B wins 624 and loses 608 against C with 64 tied, and C wins 624 and loses 608 against A with 64 tied. Only the result for A against B changes direction, and the loop becomes the order B, C, A. That is an exhaustive check of those dice at that number of throws. It is not a claim about three throws or about other dice.

Everything above is about three particular dice, and the loop is not typical of dice. Ask the census: among all the dice that can be made from the faces 1 to 6, most sets of three have a strict order (about 91 percent with one throw each), about 8 percent contain a pair that wins as often as it loses, and cycles are about a quarter of one percent. The census is exact for those dice and no others. It says nothing about dice with more sides, other faces, or a fixed total of faces.

There is real mathematics on how common intransitive dice are when the dice get large. Conrey, Gabbard, Grant, Liu and Morrison studied random dice with a computer in 2016 and conjectured that, as the number of sides grows, ties become rare and about a quarter of random triples are intransitive. Later work, including a Polymath project, has settled versions of this for particular models of random dice. This page cites that work and does not reproduce or test it, and its small, fixed-size census should not be read as evidence about the limit.

The famous set is Efron's four dice, not the three here, and the two-throw behaviour of one set says nothing about another. Whether throwing more than twice keeps rearranging the order, and for which dice, is not something this page claims to know.

Read this next to The Coin Bet You Can Always Beat. It is the same "no best choice" idea with a coin in place of dice. Two players each bet on a pattern of three flips, and one coin is flipped until a bet turns up. Whatever the first player bets, the second can answer with a pattern that wins at least 2 games in 3, and four of the eight patterns beat each other round a ring, as A, B and C do here.