Here is another puzzle I like which is in the spirit of the question. 100 people play a game as follows. Each person secretly writes a number between 1 and 1000000. The numbers are then all revealed and the person who is closest to 2/3 of the average wins a prize. If there is a tie, the prize is shared.

What number should one write down?

Now, it is clearly foolish to write down any number greater than 666667, since 2/3 of the average cannot be more than 666667. But now we can view the game as being played on the interval from 1 to 666667 instead of from 1 to 1000000. Now we can iterate again and conclude that it is foolish to choose any number greater than 444444. Ultimately (but this requires many iterations of knowledge), the only rational choice is for all players to choose 1 and to split the prize.

Here is another puzzle I like which is in the spirit of the question. 100 people play a game as follows. Each person secretly writes a number between 1 and 1000000. The numbers are then all revealed and the person who is closest to 2/3 of the average wins a prize. If there is a tie, the prize is shared between the winners.

What number should one write down?

Now, it is clearly foolish to write down any number greater than 666667, since 2/3 of the average cannot be more than 666667. But now we can view the game as being played on the interval from 1 to 666667 instead of from 1 to 1000000. Now we can iterate again and conclude that it is foolish to choose any number greater than 444444. Ultimately (but this requires many iterations of knowledge), the only rational choice is for all players to choose 1 and to split the prize.