Two players, A and B, both have the number 12 written on their foreheads.

Each one sees the number on the forehead of the other one, but does not know his own number.

A third person, C, tells them that the sum of their two numbers is either 24 or 27 and that it concerns only two positive integer numbers greater than zero.

Then C asks again and again alternating A and B whether they can determine the number on their foreheads.

A: No.
B: No.
A: No.
B: No.
A: No.

After how many “no"s, can one figure out his own number? Or there is no way, after all?

In this case, B can say Yes on his 4th turn.

Just to simplify the explanation, let's place a 4th person D in the room, who can see neither A nor B's foreheads.

When A says 'No' initially, D (and everybody else) knows that B is between 1 and 23.  If B was 24 or greater, then A could deduce that the total was not 24.

When B then says 'No' initially, then D (and everybody else) knows that A is between 4 and 23.  If A was 3 or less, then B (knowing his number to be between 1 and 23) could deduce that the total was not 27.  If A was 24 or greater, then B could deduce that the total was not 24.

When A says 'No' on his 2nd turn, then D (and everybody else) knows that B is between 4 and 20.  If B was 3 or less, then A (knowing his number to be between 4 and 23) could deduce that the total was not 27.  If B was 21 or greater, then A (knowing his number to be between 4 and 23) could deduce that the total was not 24.

Using the same logic, every Negative statement narrows the range by another 3. 

When B says 'No' on his 2nd turn, then D (and everybody else) knows that A is between 7 and 20.

When A says 'No' on his 3rd turn, then D (and everybody else) knows that B is between 7 and 17.

When B says 'No' on his 3rd turn, then D (and everybody else) knows that A is between 10 and 17.

When A says 'No' on his 4th turn, then D (and everybody else) knows that B is between 10 and 14.

At this point, B knows that the total is not 27, because he knows that he is between 10 and 14, and A's number is 12.  B says 'Yes' on his 4th turn.

Had B said 'No' on his 4th turn, then D would have concluded that A's value was 13 or 14.

Posted by Steve Herman on 2008-12-17 21:11:14