# Math Help - Card Trick!

1. ## Card Trick!

Using a 52-card deck, have three people each select a card without showing it to you. Tell them to memorize their card.

Deal one pile of 10 cards face down. Next to it deal a pile of 15 cards, and next to that deal another 15-card pile. Keep the remaining 9 cards in your hand. Have the first person put his (or her) card on top of the 10-card pile, cut as many cards as he wants from the second pile, and put them on his card. Have the second person put her card on the second pile, cut as many cards as she wants from the third pile, and put them on top of her card. Have the third person put his card on top of the third pile, hand him the 9 cards you're holding, and have him place them on top of his card.

Pick up the last pile, put it on the middle pile, and put both on the first pile. Make clear that the cards are now lost and you will find them. Take four cards off the top and place them on the bottom of the deck. Explain that you are going to flip a card up and next to it one down and keep on repeating this until you don't have cards in your hand. Tell the spectators to say "Stop" if they see their card. Deal the cards alternately into two piles, one face up and one face down, starting with the face-up pile. When all the cards have been dealt (the spectators won't see their card unless you mess up), push the face-up pile aside and pick up the other pile. Deal it into two piles in exactly the same way. Keep repeating this until you have only three cards left face down. Turn them over, and there are their cards. The top one is the third person's card, the next is the second person's card, and the bottom one is the first person's card.

Video of trick:

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Anyone want to have a crack at explaining this one? I'm interested to see how it might work.

2. Hello, mrbuttersworth!

I have it figured out!
Now I hope I can explain it . . .

Using a 52-card deck, have three people each select a card
without showing it to you. Tell them to memorize their card.
Suppose the three cards are: ${\color{red}x},\,{\color{red}y},\,{\color{red}z}.$

Deal one pile of 10 cards face down. Next to it deal a pile of 15 cards,
and next to that deal another 15-card pile.
Keep the remaining 9 cards in your hand.
The layout looks like this: . $\boxed{10}\quad\boxed{15}\quad\boxed{15}\qquad \boxed{9}$

Have the first person put his card on top of the 10-card pile, cut as many
cards as he wants from the second pile, and put them on his card.
Suppose he moves $a$ cards from the second pile to the first.

Then we have: . $\begin{array}{ccc} \\ \boxed{a} \\
\boxed{{\color{red}x}} \\
\boxed{10} & \boxed{15-a} & \boxed{15} \end{array}$

Have the second person put her card on the second pile, cut as many
cards as she wants from the third pile, and put them on top of her card.
Suppose she moves $b$ cards from the third pile to the second.

Then we have: . $\begin{array}{ccc}\boxed{a} & \boxed{b} \\
\boxed{{\color{red}x}} & \boxed{{\color{red}y}} \\
\boxed{10} & \boxed{15-a} & \boxed{15-b} \end{array}$

Have the third person put his card on top of the third pile, hand him
the 9 cards you're holding, and have him place them on top of his card.
We have: . $\begin{array}{ccc} \boxed{a} & \boxed{b} & \boxed{9} \\ \boxed{{\color{red}x}} & \boxed{{\color{red}y}} & \boxed{{\color{red}z}} \\ \boxed{10} & \boxed{15-a} & \boxed{15-b} \end{array}$

Pick up the last pile, put it on the middle pile, and put both on the first pile.
We have: . $\begin{array}{c}\boxed{9} \\ \boxed{{\color{red}z}} \\ \boxed{15-b} \\ \boxed{b} \\ \boxed{{\color{red}y}} \\ \boxed{15-a} \\ \boxed{a} \\ \boxed{{\color{red}x}} \\ \boxed{10} \end{array} \qquad\Rightarrow\qquad \begin{array}{c}\boxed{9} \\ \boxed{{\color{red}z}} \\ \boxed{15} \\ \boxed{{\color{red}y}} \\ \boxed{15} \\ \boxed{{\color{red}x}} \\ \boxed{10} \end{array}$

Take four cards off the top and place them on the bottom of the deck.
We have: . $\begin{array}{c}\boxed{5} \\ \boxed{{\color{red}z}} \\ \boxed{15} \\ \boxed{{\color{red}y}} \\ \boxed{15} \\ \boxed{{\color{red}x}} \\ \boxed{14} \end{array}$

Explain that you will turn a card face up and next to it one face down and continue
this way through the entire deck. Tell them to say "Stop" if they see their card.

Deal the cards alternately into two piles, one face up and one face down, starting
with the face-up pile. (They won't see their card unless you mess up).
$\begin{array}{cccccccccccc}\text{Face up} & 1 & 3 & 5 & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \text{Face down} & 2 & 4 & {\color{red}z} & 2 & 4 & 6 & 8 & 10 & 12 & 14 & {\color{red}y} \end{array}$ . $\begin{array}{cccccccc}1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ 2 & 4 & 6 & 8 & 10 & 12 & 14 & {\color{red}x}\end{array}$ . $\begin{array}{ccccccc}1 & 3 & 5 & 7 & 9 & 11 & 13 \\ 23 & 4 & 6 & 8 & 10 & 12 & 14 \end{array}$

Push the face-up pile aside and pick up the other pile.
Note: since this pile was dealt face-down, the order of the cards is reversed.

The pile (from top to bottom): . $\text{7 cards},\;{\color{red}x},\;\text{7 cards},\;{\color{red}y},\;\text{7 cards},\;{\color{red}z},\;\text{2 cards}$

Deal it into two piles in exactly the same way.
$\begin{array}{cccccccccccccc}\text{Face up} & 1 & 3 & 5 & 7 & 1 & 3 & 5 & 7 & 1 & 3 & 5 & 7 & 1 \\ \text{Face down} & 2 & 4 & 6 & {\color{red}x} & 2 & 4 & 6 & {\color{red}y} & 2 & 4 & 6 & {\color{red}z} & 2\end{array}$

The face-down pile is reversed.

The pile (from top to bottom): . $\text{1 card},\; {\color{red}z},\;\text{3 cards},\;{\color{red}y},\;\text{3 cards},\;{\color{red}x},\;\text{3 cards}$

Keep repeating this until you have only three cards left face down.
$\begin{array}{cccccccc}\text{Face up} & 1 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{Face down} & {\color{red}z} & 2 & {\color{red}y} & 2 & {\color{red}x} & 2\end{array}$

The face-down pile is reversed.
The cards (from top to bottom): . $\text{1 card},\;{\color{red}x},\;\text{1 card},\;{\color{red}y},\;\text{1 card},\;{\color{red}z}$

$\begin{array}{ccccc}\text{Face up} & 1 & 1 & 1 \\ \text{Face down} & {\color{red}x} & {\color{red}y} & {\color{red}z} & {\color{blue}\Leftarrow\;\text{There!}}\end{array}$

3. Congrats Soroban. In the solution you can see the "magic" the whole 15-a; 15-b is completely irrelevant, seeing as the piles will invariably be added together, gives people the illusion though. Makes you wonder, why not line the trick up like:

14 15 15 5

as opposed to taking four of the top cards and putting them at the bottom. Anyways, neat trick, great explanation. Thanks for helping my curiosity.