Donald E. Knuth's Magic Bingo Cards. Help wanted!
Effect: The magician and a spectator each select a bingo card. The numbers 1 through 6 are randomly drawn without replacement, as in standard bingo. If a called number is on a card, it is marked with a coin. The first player to complete a HORIZONTAL row wins.
Secret: The game is intransitive. The spectator picks his card first, and then the magician picks a card. According to Martin Gardner, card A beats card B,card B beats Card C, card C beats card D, and card D beats card A! So, A beats B beats C beats D beats A. If, for example,a 12 game match is played, the magician will win the majority of the 12 games. The intransitive nature of the cards is hidden, as opposed to intransitive dice. ( On dice, you can see if one die has higher numbers than the other. But these bingo cards, ????) Also, in actual performance, the cards would be secretly marked, so you know which card beats the other. Make sure cards are numbered exactly as shown.
The problem is: What percentage advantage does card A have over card B? What percentage advantage does card B have over card C? And so on. These percentages are not stated in the article.
If there is a computer programmer out there who could do a simulation on this, please post the results here.
NOTE: Reader Al Stanger wrote to me and stated that the way I have lettered the cards, A beats B, B beats D, D beats C, C beats A. In each of those four pairings, the better card wins with probability 7/15, loses with probability 13/30, and ties with probability 1/10.
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