Card Guessing

Question

Consider there are 4 blue and 4 red cards. Host gets 2 cards randomly, no one knows what cards they are. Then he places 2 random cards at the forehead of each player A, B and C. Players do not know the color of their own cards, but know those of other players. They have to guess what cards they have. A said he do not know, B said he do not know, then C said he do not know. Then A said he knows now. Please explain the logic how did A get it.

Solution

First, we to think about the situations of all players. In what condition a player knows or does not know his own card? There are 2 of them.

1. A must have Red–Red if all cards of B and C are Blue (B has Blue–Blue and C has Blue–Blue).
2. A must have Blue–Blue if all cards of B and C are Red (B has Red–Red and C has Red–Red).
3. A won’t know what cards he has if cards of B and C have at least one different color (all other situations beside 1 and 2).

Repeat this logic to players A, B and C. Since all of them don’t know what cards they have (situation 3), this implies all players has Red–Blue cards. Therefore A knows he has Red–Blue cards.