The Antisocial Club

The first person must take either stool 9 or 17 (because of symmetry, it doesn't matter which). Assume they pick seat 9. The next person will pick seat 25, since it is the furthest from seat 9. The next two people will take Seats one and 17. The next three will occupy 5, 13, and 21. The next six will occupy 3, 7, 11, 15, 19, and 23. This seats the maximum of 13 people, and no one is sitting next to another person. If a seat other than 9 or 17 is chosen first, the total bar patrons will be less than 13.

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Stamps

B says: "Suppose I have red-red. A would have said on her second turn: 'I see that B has red-red. If I also have red-red, then all four reds would be used, and C would have realized that she had green-green. But C didn't, so I don't have red-red. Suppose I have green-green. In that case, C would have realized that if she had red-red, I would have seen four reds and I would have answered that I had green-green on my first turn. On the other hand, if she also has green-green [we assume that A can see C; this line is only for completeness], then B would have seen four greens and she would have answered that she had two reds. So C would have realized that, if I have green-green and B has red-red, and if neither of us answered on our first turn, then she must have green-red. "'But she didn't. So I can't have green-green either, and if I can't have green-green or red-red, then I must have green-red.' ...

Suicidal Monks

If there is only one monk with red eyes, then he sees all the others are brown-eyed, so he must be the red-eyed one. He kills himself the first night. If there are two monks with red eyes, then each sees one monk with red eyes and reasons that if this other monk is the only monk with red eyes, he will kill himself the first night. Neither monk kills himself the first night, so they each reason that they must have red eyes too. Both kill themselves the second night. If there are three, each expects the other two to commit suicide the second night. This doesn't happen, so each deducts that he must be a third, and the suicides happen the third night. Extends to four, five, etc. If the suicides happened n midnights after the tourist's remark, then there are n monks with red eyes.

Backwards Clock

Suppose that a second pair of hands turns together with the wrong pair of hands, but then in the correct way. When the wrong pair is in the same position as the correct pair, this means that the time is shown in the right way. First look at the hour-hands that are at the twelve. One turns with the correct speed, the other with a speed that is twelve times as small. These two hands are in the same position again when the 'slow' hand has progressed x minutes. The fast hand then has progressed 60+x minutes. For the time x that passed, then holds: (60+x)/12 = x. This means that x = 5 5/11 minutes. For the minutes-hands that start at six holds the same. The confused clock therefore shows the correct time again at 5 5/11 minutes past 7.

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