Splitting a deck in the dark

Question

You are in a completely dark room with a deck of 52 cards. You are told that exactly 10 of them are face up and the rest face down, in unknown positions. You cannot see or feel which cards are face up. Divide the deck into two piles so that each pile contains the same number of face-up cards. The piles need not be the same size, and you may turn cards over.

Accepted Answer

Count off any 10 cards into a separate pile and turn every one of them over. The two piles now have equal numbers of face-up cards. Why. Suppose the 10 cards you took contain k face-up cards (you do not know k). Then the remaining 42 cards contain the other 10 minus k face-up cards. Your pile of 10 has k face-up and 10 minus k face-down. When you flip all 10, the face-down ones become face-up: your pile now has 10 minus k face-up cards. That is exactly the number in the other pile. Check with an example. If your 10 cards happen to include 3 of the face-up ones, the big pile has 7 face-up. Aft…

Splitting a deck in the dark

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