Let’s Crack a Magic Trick

Magic is a great entertaining and fascinating art. There are several kinds of magic. Magic usually uses sleight of hand, misdirection, lying (of course), magic props, pre-show work, secret assistant, mathematics etc. A long list of magic types can be seen here. I have always been fascinated by both mathematics and magic for a long time. I’m also a kind of person who likes to destroy the fun in a magic trick trying to figure that out. (Kind of conflicting characteristics, I know). In this post, I’ll be teaching you a very simple card trick. Then discuss the internal mechanism of the magic trick using a mathematical model.

Let’s do the trick

Let’s call the trick “Clubs, Hearts and Diamonds trick”. Since the trick uses these kinds of cards and the exact words when performing. You can watch this youtube video to learn the trick under 4 minutes. I’ll list the steps below to make things clear.

To perform this trick, we need 3 cards from Clubs, Hearts and Diamonds. Let’s choose J, K, Q cards from each kind. Altogether, 9 cards are used in the trick.

1. Make 3 piles such that each pile contains cards from the same kind.
2. Ask your spectator(s) to select any pile.
3. Deal the 3 cards in that selected pile in 3 separate places.
4. Again ask your spectator(s) to select another pile of remaining two piles.
5. Again deal the 3 cards in that selected pile on top of the previously dealt places. The order is irreverent.
6. Also, take the last remaining pile and deal the 3 cards as did in step 5.
7. Now ask the spectator(s) to collect the new 3 piles in any order he/she likes.
8. Then, ask the spectator(s) to cut the 9 card pile anywhere he(she) wants. You can let any number of cuts to be performed.
9. Optional: Shuffle the 9 card pile by dealing the cards in two separate piles in an alternating manner and collect them in any order.
10. Let’s say rotating the card pile by n times is taking a card from the top of the pile and placing it in the bottom, and repeating it n times.
11. Now, rotate the 9 card pile 3 times. Count the letters in the word “CLUB” while rotating. As per the letter ‘B’, take the card out and place it in a separate place. (Let’s call this P1)
12. Rotate the remaining 8 card pile 4 times with the word “HEART”. As per letter ‘T’, take the card out and place it in a separate place than the previous place. (Let’s call this P2)
13. Rotate the remaining 7 card pile 6 times with the word “DIAMOND”. As per the last letter ‘D’, take the card out and place it in a separate place than the previous two places. (Let’s call this P3)
14. Now Deal the remaining cards as in the steps 11, 12 and 13 but place them in the places P3, P2 and P1 instead of placing them in P1, P2 and P3.
15. Then there will be 3 cards remaining. Deal those cards without any rotation on the places, P3, P2 and P1.
16. Now reveal that in each pile P1, P2 and P3, there are only cards from the same kind.

What’s Happening

In the very beginning of this post, I stated that mathematics is used in some magic tricks. What I mean by that is inside the magic trick, some tautology is being used. Tautology is a logic statement, which is always true.

Mathematical equivalences are always true statements. For example,

(x+y)² ≡ x² + 2xy + y²

((p+ 5 - 3)/2 + 1) × 2 ≡ p

In childhood, we were used to playing games using these equivalences. There is somewhat similar (considering the algorithm of the trick) happening in the magic trick. Let’s crack the trick.

Understanding the presented trick

This trick only uses a tautology. To understand the trick, we have to first model the trick in a mathematical model. Then perform the steps on the model. The complications of having the cards face down and unnecessary moves are making the procedure a real magic trick. Since we are going to crack (reveal or understand) the trick, let’s go with the mathematical model, step by step.

Let’s call the three kinds of cards as A, B and C (our notation for the model). Since their order is not considered, no need to worry which letter is assigned to which kind. Let’s perform the 16 steps of the tricks by using this model.

Step 1: Place 3 different kinds of cards in 3 different piles, facedown.

Steps 2–6: Note that since A, B, C could be any of the kinds, we suppose the spectators choose the kinds of A, B and C in the order to make the 3 piles.

Step 7: Since 3 piles are identical(According to the model), the spectator’s choice is un-important in collecting the piles.

Step 8: Cutting the pile. Note that cutting doesn’t change the order of a pile of cards. It only rotates the card pile and changes the starting point. Note that our notation for the different cards is just A, B, C, let’s again re-notate the cards to be in the positions of the above figure.

Step 9: This step also doesn’t change the order of the 9 card pile. See the following figure. Note that we’ll have to re-notate as A, B, C ← B, A, C to refer the above figure’s pile.

Step 11: Rotating with the word “CLUB”. Let’s use a sequence of letters to represent the pile of cards.

Note that the notation is,
Top → A B C A B C A B C ← Bottom

Initial pile: A B C A B C A B C

After rotate(3): A B C A B C A B C

After dealing: B C A B C A B C | P1={A}

Step 12: Rotating with the word “HEART”.

Initial status: B C A B C A B C | P1={A}

After rotate(4): C A B C B C A B

After dealing: A B C B C A B | P1={A}, P2={C}

Step 13: Rotating with the word “DIAMOND”.

Initial status: A B C B C A B | P1={A}, P2={C}

After rotate(6): B A B C B C A

After dealing: A B C B C A | P1={A}, P2={C}, P3 = {B}

Step 14: Rotating for the second time deal.

Initial status: A B C B C A | P1={A}, P2={C}, P3 = {B}

After rotate(3) with “CLUB”: B C A A B C

After dealing: C A A B C | P1={A}, P2={C}, P3 = {B, B}

After rotate(4) with “HEART”: C C A A B

After dealing: C A A B | P1={A}, P2={C, C}, P3 = {B, B}

After rotate(6) with “DIAMOND”: A B C A

After dealing: B C A | P1={A, A}, P2={C, C}, P3 = {B, B}

Step 15: After dealing the rest, the 3 piles will be as follows. Note that the dealing order is P3, P2 and then P1.

P1={A, A, A}, P2={C, C, C}, P3 = {B, B, B}

Step 16: Voila! It’s no more magic !!!

Finally…

The choice of the rotation terms “CLUB”, “HEART” and “DIAMOND” is a careful selection to suit the trick. Note that these are not the only terms which can be used with the same trick. You can try to invent a trick involving all 4 kinds and the same procedure. The rotation terms should be then carefully chosen.

I hope you enjoyed the trick and the mechanism behind it. Now you enjoy one less magic trick !!!