By Mac Lane, Birkhoff, Delorme, Lavit, Mezard, Raoult
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Additional info for Algèbre, solutions developpees des exercices, 3eme partie, Les grands theoremes [Algebra]
How, though, should A play so as to make his wins as big as possible, and how should B play so as to lose as little as possible? Most games have numerous and complicated strategies, but here each player is limited to two: he can show one finger or he can show two. The "payoff matrix" can therefore be drawn on a 2-by-2 square as shown in Figure 4, left. By convention, A's two strategies are shown on the left and B's two strategies are shown above. The cells hold the payoffs for every combination of strategies.
Is not a multiple of 13, because 13 is prime and the factors of 12! do not include 13 or any of its multiples. But, astonishingly, the mere addition of 1 creates a number that is divisible by 13. Wilson's theorem is one of the most beautiful and important theorems in the history of number theory, even though it is not an efficient way to test primality. There are many simply expressed but difficult problems about factorials that have never been solved. No one knows, for example, if a finite or an infinite number of factorials become primes by the addition of 1, or even how many become squares by the addition of 1.
If B does not have the jack, he answers no. This places him in a quandary, although one that proves to be short-lived. If he thinks A is not bluffing, he calls the jack and ends the game, winning if his suspicion is correct. If he does not call it and the hidden card is the jack, then A (who originally asked about it) will surely call the jack on his next play, for he will know with certainty that it is the hidden card. Therefore, if A does not call the jack on his next play, it means he had previously bluffed and has the jack in his hand.