Stamps

[jason]

During a game show, the host has placed three contestants (A, B, C) in a closed room. The host places one stamp on each of the contestant’s forehead in such a way that each contestant can only see the other stamps, but not his own. He places a white or black stamp to the forehead of A, a red or green stamp to B and a yellow or blue stamp to C. He then asks each of them to simultaneously write on a piece of paper the color of the stamp they think that they have on their own forehead. The only choices that are permitted are either a color or "I don't know". The contestants will win if at least one of them correctly guesses his own color, but with the restriction that no one must give a wrong answer (for example, if 2 of the 3 contestants reply "I don't know" and the 3rd one guesses correctly, they win the contest).
The host allows the contestants to discuss and coordinate, in order to decide a strategy. What is the optimum strategy that will give them the biggest chance to win?

[Guest]

I'm not sure the question is right or that I have understood it properly, as there are a lot of bits to the question which do not make a difference.

As it stands the optimal strategy is for A, B, C to guess white, don't know, don't know, this gives a probability of 1/2 of winning.

If anyone doesn't choose "don't know" then their guess has a 50% chance of being wrong (there is no information that can be gained about your stamp from looking at other stamps), so there is always at least a 50% chance the overall game is lost. So the strategy I gave is the best you can hope to do.

Hope this helped,

R. Baber.

[jason]

I changed the phrase in bold to "no one must give a wrong answer" (I had written "no one must NOT give").

[Guest]

This is a variation of the Ebert's hat game, which is solved by using Hamming codes.