There is a standard deck of cards, with some cards face-up and the rest of the cards face-down. You are told exactly how many cards are face-up, but you are not allowed to look at the cards.
Without seeing which cards are face-up and which are face-down, how can you divide the deck into two piles of cards that contain the same number of face-up cards?
Hint
The two piles of cards can have different numbers of cards in them – they just need to have the same number of face-up cards.
Solution
Let’s say you are told n cards are face-up. Take any n cards from the deck to form a new pile, and flip all of those cards over. The remaining pile of cards in the deck will have the same number of face-up cards as the new pile!
How can this be correct? Well when you make a new pile using n cards from the deck, some number of those cards will be face-up – let’s call that m face-up cards in the new pile.
Since the deck started with n face-up cards, the remaining pile of cards in the deck must have n – m face-up cards.
Now when you flip the cards in the new pile over, all m of the face-up cards will become face-down, and the remaining n – m face-down cards will become face-up. Therefore both piles will have n – m face-up cards!