Sam and Pete, two perfectly intelligent logicians, are sitting facing each other, each with a hat on. The number on Pete's hat, they are told, is the sum of two integers larger than 1; the number on Sam's hat is the product of these two integers. The following conversation ensues:

Sam: I don't know the number on my hat.
Pete: I don't know the number on my hat.
Sam: I don't know the number on my hat.
Pete: I don't know the number on my hat.
Sam: I don't know the number on my hat.
Pete: I don't know the number on my hat.
Sam: OK, now I know the number on my hat.
Pete: And I know the number on mine.

What are the numbers on Pete's and Sam's hats?

Ok, so Sam knows the Sum and Pete knows the Product (handy!)

Since Sam doesn't know the integers on his first turn, that means there isn't one unique pair that add to the sum he is seeing. This only eliminates 4 (2+2) and 5 (2+3).

Since Pete doesn't know the integers on his first turn, that means the product is not a product of two primes. Also, the product is not a cubed prime (because the only factors can be P and P^2), but this doesn't really end up coming into play, so I'm only going to focus on pairs of primes. Also, I'm only going to focus on integers from 2-20 (originally I looked at a much larger range of numbers, but when I found the solution I saw that focussing on 2-20 would suffice). There are only 8 primes in this range, so that means we can eliminate 1+2+3+4+5+6+7+8 = 8*9/2 = 36 integer pairs (note that two of them were the two that Sam eliminated on his first turn). So we've eliminated 36 integer pairs so far.

Knowing the integer pairs Pete eliminated so far, Sam still doesn't know what the answer is yet. That means there still isn't one unique pair that add to the sum he is seeing. This only eliminates 7 (4+3).

Knowing the integer pair that Sam just eliminated, Pete still doesn't know what the answer is yet. That means there still isn't one unique pair that multiply to the product he is seeing. This only eliminates 12 (6*2).

Knowing the integer pair that Pete just eliminated, Sam still doesn't know what the answer is yet. That means there still isn't one unique pair that add to the sum he is seeing. This only eliminates 8 (4+4).

Knowing the integer pair that Sam just eliminated, Pete still doesn't know what the answer is yet. That means there still isn't one unique pair that multiply to the product he is seeing. This only eliminates 16 (8*2).

Now Sam knows the answer. That means the sum he is looking at can only be made by one of the remaining integer pairs, and that is 10 (6+4)

Therefore the Product on Sam's hat is 24 and the Sum on Pete's hat is 10, and the integers are 6 and 4.

Edited on November 7, 2006, 6:05 pm

Posted by nikki on 2006-11-07 18:02:12