Two logicians place cards on their foreheads so that what is written on the card is visible only to the other logician. Consecutive positive integers have been written on the cards. The following conversation ensues:
A: "I don't know my number."
B: "I don't know my number."
A: "I don't know my number."
B: "I don't know my number."
........ n statements of ignorance later..........
A or B: "I know my number."
What is on the card and how does the logician know it?
I am assuming that both logicians know that the numbers are consecutive positive integers, and they both know that the other is going to think the problem out correctly.
If the first logician saw a 1, then he would know that his number was two (as zero is not positive).
If the second logician then saw a 1, he would similarly know he had a 2. If he saw a two, then, he would know that his number was 3 (if it was 1, the first logician would have been able to correctly guess a 2).
If the first logician, then, saw a 3, he would know that his number was a 4; like above, if his number was 2 the other logician would have correctly deduced a 3 on his own head.
Going on as such, after n 'statements of ignorance,' the logician who sees the number n+1 on the other's forehead will be able to deduce that his own card bears the number n+2.

Posted by DJ
on 20030411 05:50:45 