Puzzle 73 | The Card Game

A blind man is handed a deck of 52 cards and told that exactly 10 of these cards are facing up. How can he divide the cards into two piles, not necessarily of equal size, with each pile having the same number of cards facing up?

Solution:
If the original pile has c cards with f cards facing up, then the blind man divides them into piles of size f and c – f. Then he flips all cards in the pile with f cards. Let’s see why it works for c = 52 and f = 10. The blind man would divide the cards into two piles with 10 and 42 cards each. If there are k face-up cards in the 10-card pile, then there must be 10 – k face-up cards in the 42-card pile (because the total number of face-up cards is 10). So by flipping all cards in the 10-card pile, the number of face-up cards in both piles would become equal to 10 – k.

This puzzle is contributed by Praveer Satyam.