There are 3 people Abel, Bill and Clark. Three of them have numbers (positive integers) written on their hats. They can see the numbers on the other two hats but not their own. The number on one hat is the sum of the numbers on the other two hats. They are given this information and asked in turn if they can identify their number.
In the first round Abel, Bill and Clark each in turn say they don’t know. In the second round Abel is first to go and states his number is 50. What numbers are on Bill and Clark?
Abel has 50, Bill has 20 and Clark has 30.
Abel on his first turn obviously doesn’t know whether his number is 50 or 10. Similarly neither Bill
nor Clark can immediately figure out their numbers. However, on his second turn Abel can reason:
If mine is a 10, then Clark would know his number is either 10 or 30.
If it is 10, Bill would immediately know his number is 20. But he didn’t know.
So Clark should know his number is 30. Now since Clark didn’t know, my number must be 50.
With this kind of reasoning we can also rule out all other combinations. So [50, 20, 30] is the only solution to this puzzle.