Edmund's Brainteasers
Problem 21
Solution
Problem 21: Another hat problem. Solution.
Suppose that Jane saw one of the numbers 1, 2 or 3 on Phil's hat. She would then know that number on her hat was 3 more than that on Phil's hat (it couldn't be 3 less) so she would stand up after the first ring of the bell. If Jane didn't stand up after the first ring of the bell, Phil would know that he doesn't have 1, 2 or 3 on his hat so, if he sees 4, 5 or 6 on Jane's hat, he would stand up after the second ring of the bell. If neither has stood up after 2 rings of the bell both know that nobody has 1, 2, 3, 4, 5 or 6 on their hat so if either saw 7, 8 or 9 on the other's hat they would stand up after the third ring. We can continue to argue in this way. We are told that Jane sees that the number 25 on Phil's hat. Neither will stand up after the first 7 rings of the bell. If Jane has 22 on her hat then Phil will know when Jane doesn't get up on the seventh ring that his hat must have 3 more so his hat must be 25 and Phil wins getting up after the eight ring. If Jane has 28 on her hat then she will win by getting up after the ninth ring.