Not Sweating the Small Stuff

'The Three Prisoners' – A 1950's CIA IQ Entrance Test

The Three Prisoners Relaxing in the Goal's Music Room - Before They Hear about the Execution Idea!


THE PROBLEM

The goal cell holds three prisoners. In the morning two are to die. The jailer is to decide which one is to live.

In view of the prisoners, our custodian takes five small squares of paper and on three he places a cross and on two a circle. He then tells the prisoners he will randomly select three of the pieces of paper and discard the remaining two. And that he will place one square to each of the prisoners’ foreheads. Each will be able to see what is on the others’ foreheads, but not his own. In fact the jailer secretly contrives to select only the three squares with crosses on them.

So each prisoner now has a square with a cross on his forehead. But while each sees a cross on the forehead of each of the other prisoners, he himself could in fact have a circle or a square attached to his own.

The jailer tells the prisoners that the one to tell what is on his forehead with the right reasoning will be freed. A guess, even if right, with the wrong logic will result in death.

One prisoner took less than five minutes to determine the answer. And gain his freedom.

What was his argument?

THE SOLUTION

Let’s call the prisoners A, B and C, with A being the one who is freed.

A would say to himself 'If I were in fact B and saw a circle on A’s forehead and a cross on C’s, I (as B) would know I couldn’t have a circle because C would have seen two circles (one on A and one on me, B) and gone to the jailer and said he must have a cross because there are only two circles'. Thus, if B saw a circle on A's forehead he would use this logic to determine that he must have a cross. And so gained his (B's) exit. But because B didn’t use this logic, B must not see a circle on A’s forehead, but a cross. Thus A argued for his cross and gained his escape from prison!