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Mathemagical Card Trick

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Yesterday somebody showed me a mathematical card trick and asked for an explanation. It's a simple solution, but not a very obvious one. This might be fun for math teachers to share with their students. All you need is a normal deck of cards and some very basic algebra skills. Start with a full deck of 52 cards in your hand. Put a random card face up. The value of the card is a number from 1 to 13 (Ace is 1, King is 13). Subtract the value of the card from 13, and then that number of cards (all face up) on top of the card . So, if you put down a King, that's 13. Subtract 13 from 13 to get zero, so you don't add any cards on top of the King. If the card was a seven, you subtract (13 - 7 = 6); so you add six cards face up on top of the seven. Repeat step 2, making as many additional piles as you can. If you end up putting a card down but cannot complete the step, then pick up that pile and keep it in your hand.  Choose three of the piles to keep on the table and add the rest