Math Card Trick

Eric Shrader

My grandfather taught me this simple trick when I was young:

Deal out 27 cards face up into a grid of 9 rows and 3 columns. Do this by dealing out 3 cards horizontally in a row, then 3 more cards just below the first 3, then 3 more, etc., until you have 9 rows. (It's best to overlap cards in a column so that the columns aren't so long; just make sure that the values of all cards are visible.) Discard the remaining cards. Only these 27 will be used to play.

Ask a spectator to mentally pick a card and remember it. Ask him to tell you only which of the 3 columns it is in.

Collect the 27 cards into a deck. Gather them vertically such that the column containing the spectator's card is second. Pick up cards from the top of the column to the bottom, keeping them in the same order. For example, if the spectator tells you column 1, then first gather column 2 or 3, then column 1, and then the remaining column. When you're done, the top card of the deck should now be the top card of the column you collected first, followed by the rest of that column in order. Then the 10th card of the deck will be the card at the top of the column containing the spectator's card, etc. If you overlapped the cards as suggested, gathering and keeping them in the correct order is easy.

Now deal the cards again. Deal exactly as described above - horizontally across the rows first.

Again, ask him to tell you only which column contains the card.

Pick up the columns vertically - exactly as above - making sure that the column containing the card is picked up second.

Finally deal them out a third time in exactly the same way, ask which column contains the card, and gather them up in the same manner.

The spectator's card will now be the 14th one in the deck. To add drama, I usually deal out the cards one at a time face down. I don't make it obvious that I'm counting and I don't look at any of the cards. Instead, I'll hesitate over certain cards, pretending to get a "vibe" from them, and then I'll finally settle on the right one.

This seemed like pure magic when I first learned it. Only when I got older did I realize that it's actually simple math. Do you see how it works?

Subject:Math Card Trick - Better Version
Date:Sun, 2 Nov 2003 21:58:05 -0000
From:Keith Riding

Re the math card trick - never mind number 14 - my grandfather taught me how to make the card come out at any number the subject chooses.

27 cards dealt just as you say - 3 times. If person chooses say number 15, then :-

  1. 1st time round count in 3's viz. 1 2 3: 4 5 6: 7 8 9: etc til get to 15. ie to '13 14 15'. As seen the 15 is 3rd of the 3 therefore is put in 3rd ie bottom.

  2. 2nd time round count in groups of "3's", vis 1 2 3 counts as '1'. 4 5 6 counts as '2' and 7 8 9 counts as '3'. Continuing round again 10 11 12 counts as '1'. '13 14 15' counts as '2', therefore this time put the pile chosen in 2nd ie middle

  3. 3rd time round is obvious - if number 1 to 9 put on top, 10 - 18 put in centre, 19 to 27 put at bottom. For number 15 therefore put in centre again.

Using above rules, card will end up at whatever number position is chosen.

Give it a try.

Keith Riding

The following books may be of interest to those who liked the card trick above. Besides other math curiosities, each describes additional math tricks with playing cards. [Rouse Ball, p. 328-329] attributes the trick to M. Gergonne who in 1813-1814 proved a generalization that dealt with NN cards arranged in N rows of NN-1 cards each. (An interactive demonstration is available elsewhere.)

  1. M. Gardner, Wheels, Life and Other Mathematical Amusements, W. H. Freeman and Co, 10th printing, 1999
  2. M. Gardner, Mathematics, Magic and Mistery, Dover, 1956
  3. W. W. Rouse Ball, H. S. M. Coxeter, Mathematical Recreations and Essays, Dover, 1987
  4. W. Simon, Mathematical Magic, Dover, 1964

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