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 RedRed if all cards of B and C are Blue (B has BlueBlue and C has BlueBlue).
2. A must have BlueBlue if all cards of B and C are Red (B has RedRed and C has RedRed).
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 RedBlue cards. Therefore A knows he has RedBlue cards.

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