Algebra Card Trick

A parent of one of my students showed me this math card trick and asked me for an explanation of why it works. I share this with you below and ask that you work out the algebra involved in the explanation.

    The Trick
  1. Turn the top card of a regular 52 card deck face up.
  2. Begin counting from the face value of the first card, turning cards up from the deck until you have counted through fourteen (Aces have a face value of "1", Jacks "11", Queens "12", Kings "13").
    a. Example: If you turn up a nine for the first card; you would turn over 6 more cards for that pile, saying "9", "10", "11", "12", "13", "14".
  3. Next, you form three more piles in the same manner. Then turn all four piles face down
  4. Place leftover cards in a discard pile.
  5. Have a student pick up 2 of the piles and put them in the discard pile.
  6. Have a student turn up the top card of one of the piles. After counting the discard pile, you can tell the students the face value of the top card in the other pile.

    Here is the process you go through to find the face value of the card:
  1. Deal off 20 cards from the discard pile.
  2. Remove the number of cards from the discard pile that corresponds to the face value of the turned up card.
  3. The number of cards remaining in the discard pile will equal the face value of the mystery card.

    Explanation of the trick
  1. Let x = the face value of the top card on one of the remaining two piles.
  2. Let y = same as above for the second pile.
  3. How many cards are in the x-pile? (Represent this in terms of x and a number)
  4. How many cards are in the y-pile?
  5. Let c = the number of cards in the discard pile (including the removed-and-unused pile).
  6. Then c = 52 - ( _____ + _____ ) from above
  7. So c = 20 + x + y
  8. Now solve this equation for x:         x = ________
  9. Now solve the equation in step 7 for y: y = _________
  10. Looking at the equations in #8 and #9, you should see why this trick works.
    1. In #8, if you are looking for the face value of the x-pile, you would take the discard pile (c), deal off 20 cards, then deal off the number equal to the face value of the y-pile.
    2. In #9, if you are looking for the face value of the y-pile, you would take the discard pile (c), then deal off 20 cards, then deal off the number equal to the face value of the x-pile.

      Answers to the Explanation above
    • #3: 16 - x
    • #4: 16 - y
    • #6: c = 52 - (16 - x + 16 - y)
    • #8: x = c - 20 - y
    • #9: y = c - 20 - x

    Click here for Comic strip on card tricks