Mathemagical Card Trick




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.



  1. Start with a full deck of 52 cards in your hand.

  2. 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.

  3. 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. 

  4. Choose three of the piles to keep on the table and add the rest of the piles to your hand. 

  5. Turn over the three piles so they are all face down. Then turn over the top card from two of the piles. One pile must remain face down (as in the picture above).

  6. Add the values of those two top cards that you uncovered, and then subtract that number of cards from your hand.

  7. Finally, remove another ten cards from your hand.

  8. Count the number of cards left in your hand. Whatever number you get is the same number as the top card that is face down. Turn it over and see!

    Neat trick! 

    So how does it work?

First, let's identify a couple mathematical relationships.

The number of cards in your hand is equal to 52 minus the number of cards on the table. This is obvious, because there are 52 cards in the deck, and they're either in your hand or on the table. So we have: 


Formula 1.  H = 52 - T


We can represent the number of cards on the table by looking at how the piles were constructed. Each pile has a certain number of cards, which we can call P. Since there are three piles at the end, we can call them P1, P2 and P3.

          Each pile started with a card, so let's call the pile starters C1, C2 and C3. The number of cards in each pile is equal to 1 + (13 - C). That is because you start with one card and then count from C up to King, which is the 13th card. That gives us: 


Formula 2.   P = 14 - C


Using Formula 2, we can calculate the number of cards on the table--the number of the cards in the three piles--like this:

T = (14 - C1) + (14 - C2) + (14 - C3)

That is equal to:

T = 42 - C1 - C2 - C3 

            

            When there are three piles left on the table and you turn them over, the top three cards are C1, C2 and C3. We want to know the value of the hidden top card. We can choose any of the three to see how it works, so let's go with C1. That gives us:

C1 = 42 - C2 - C3 - T

42 is a nice number, because it's only 10 less than 52, and that's the total number of cards in the deck. So let's change 42 to (52 - 10), like this:

C1 = 52 - 10 - C2 - C3 - T

Now let's use formula 1, which says that the number of cards in your hand is 52 - T.  We have (52 - T), so we can substitute H to get:

C1 = H - 10 - C2 - C3


That's it! 

That last formula tells us that the value of a top card (C1) is equal to the number of cards in our hand, minus ten, and minus the values of the other two top cards. So, if you remove that many cards from your hand, you will be left with the value of the hidden card.


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