# Donald Knuth's Nontransitive Bingo Cards

In Time Travel and Other Mathematical Bewilderments, Martin Gardner presents a set of four nontransitive bingo cards designed by Donald Knuth (pp. 61). The rules are that the first player to complete a horizontal row wins. Gardner does not delve into the mathematics but merely mentions that, probabilistically, A beats B, B beats C, C beats D, and D beats A.

I was so baffled that I immediately went on to to verify those results. He was right.

Now, with dice or heads tails sequences, I can understand nontransitivity; can someone please give the mathematical explanation of how and why nontransitivity occurs in the above game.

• An observation: Where card $x$ beats card $y$, the numbers on card $x$ that are not on card $y$ are in different rows, but the numbers on card $y$ that are not on card $x$ are in the same row. This gives card $x$ the advantage, although my proof is too long to fit in this comment. – Steve Kass Feb 27 '15 at 21:55
• @SteveKass You and Fermat would have gotten along. :) – eigenchris Feb 27 '15 at 22:27
• What does it mean for one card to beat another one? Is there a payoff involved? Because clearly there are some situations where B will win before A, etc. – Ibrahim Tencer Feb 27 '15 at 22:59
• @IbrahimTencer, the numbers $1$ through $6$ are called out in random permutation. One card "beats" another if it wins more of the permutations. – Barry Cipra Feb 27 '15 at 23:05
• I changed the wording; thanks for that. – blackened Feb 27 '15 at 23:06

Let's say two players take cards $A$ and $B$ and we roll a 6-sided die to call out the numbers.
If a $1$ or a $3$ is rolled right off the start, then player $B$ cannot win using their top row and must rely on a $5,6$ victory. Player $A$ does not experience this problem if something on player $B$'s card is rolled first.