You and Your Friend
You and Your Friend played a game with a deck of cards, and the rules of the game is shown below:
-You get all the 26 black cards, and your friend gets all the 26 red cards.
-You and your friend flip the cards so that nobody can see what's under them.
-You may say any number, and swap that number of cards with your friend.
-In any point in the game, you have a chance to say that you are sure you have more red cards than your friend has black cards.
-You and your friend will open up their cards and see whether your deduction is correct.
-If your deduction is correct, you win, but if your deduction is wrong, or you do not make a deduction during the game, you lose.
Assume you and your friend played optimally, who will always win?
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1 solution
You will always have the same number of red cards as your friend has black cards.
This can be seen easily. Say you swap n cards, then you have n red cards. Therefore, since you have 2 6 total cards, you have 2 6 − n black cards. Since the total number of black cards is 2 6 , your friend has 2 6 − ( 2 6 − n ) = n black cards. So you have n red cards, and your friend has n black cards. Therefore, you have the same number of reds as your friend has blacks.