A 52 Card Conundrum

Nov 17, 2003

RAY: I'm going to hand you a deck of normal playing cards-- 52 cards and no jokers. You're going to be placed into a darkened room. You'll also be blindfolded, and you'll be naked.

TOM: Are you going to stick pokers in my eyes, too?

RAY: The deck of cards will have 13 of the 52 cards turned face up. They're all mixed up in the deck. You don't know where they are. And you can't tell.

TOM: Because I'm blindfolded.

RAY: Right. Your challenge is to arrange the cards in two piles so that each pile has the same number of cards facing up.

TOM: How can I do that? There's 13 of them.

RAY: That's your problem. But here's a hint: they don't have to be equal piles. But, if the main deck winds up with, say, 9 face-up cards, the other pile has to have 9 face-up cards, too.

Got a clue how to solve this card conundrum?


RAY: Here's the answer. I did give a hint. I said the two piles of cards you create do not have to be equal in number. In fact they can't be. Only the number of face-up cards has to be equal. So when you're all done you'll have two piles of cards, one of which will have 52 minus X and the other one will have X number of cards, and both of those piles will have an equal number of face-up cards.

TOM: The other confusing part is that there are 13 of these "face up" cards.

RAY: It doesn't have to be 13. It could be six cards face up. What you were thinking was that there had to be some way to divide the 13 face-up cards in half. So there's going to be six and a half cards, face-up cards in one pile.

TOM: Obviously not.

RAY: In fact, it's entirely possible that when you're all done there will still be 13 face-up cards in one pile in the original deck. And the new pile will have 13 face-up cards.

Here's how you do it. You're holding the deck of cards. You deal 13 cards into a new pile. So now you have one pile. You have 13 cards, and the other pile has 39 cards.

You take the 13 cards and you just dealt them to the new pile and you flip the whole pile over. You now have the same number of face-up cards in both piles.

TOM: How do I know that?

RAY: I'll give you an example. What if all 13 cards were on the top of the deck? Now you've dealt them all face up, right?

TOM: Right.

RAY: When you flip them they're now all face down. The two piles now have the same number of face-up cards- zero.

It makes no difference how many you start with, as long as you deal them out. If you know the number of face-up cards to begin with, you deal that number of cards into a separate pile and then you flip that pile over, both piles will always wind up with the same number of face-up cards.

TOM: Wow! The winner is Patrick Ivers from Laramie, Wyoming.

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