Hello, mrbuttersworth!

I have it figured out!

Now I hope I can explain it . . .

Quote:

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: $\displaystyle {\color{red}x},\,{\color{red}y},\,{\color{red}z}.$

Quote:

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: . $\displaystyle \boxed{10}\quad\boxed{15}\quad\boxed{15}\qquad \boxed{9}$

Quote:

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 $\displaystyle a$ cards from the second pile to the first.

Then we have: . $\displaystyle \begin{array}{ccc} \\ \boxed{a} \\

\boxed{{\color{red}x}} \\

\boxed{10} & \boxed{15-a} & \boxed{15} \end{array}$

Quote:

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 $\displaystyle b$ cards from the third pile to the second.

Then we have: . $\displaystyle \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}$

Quote:

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: . $\displaystyle \begin{array}{ccc} \boxed{a} & \boxed{b} & \boxed{9} \\ \boxed{{\color{red}x}} & \boxed{{\color{red}y}} & \boxed{{\color{red}z}} \\ \boxed{10} & \boxed{15-a} & \boxed{15-b} \end{array}$

Quote:

Pick up the last pile, put it on the middle pile, and put both on the first pile.

We have: . $\displaystyle \begin{array}{c}\boxed{9} \\ \boxed{{\color{red}z}} \\ \boxed{15-b} \\ \boxed{b} \\ \boxed{{\color{red}y}} \\ \boxed{15-a} \\ \boxed{a} \\ \boxed{{\color{red}x}} \\ \boxed{10} \end{array} \qquad\Rightarrow\qquad \begin{array}{c}\boxed{9} \\ \boxed{{\color{red}z}} \\ \boxed{15} \\ \boxed{{\color{red}y}} \\ \boxed{15} \\ \boxed{{\color{red}x}} \\ \boxed{10} \end{array}$

Quote:

Take four cards off the top and place them on the bottom of the deck.

We have: . $\displaystyle \begin{array}{c}\boxed{5} \\ \boxed{{\color{red}z}} \\ \boxed{15} \\ \boxed{{\color{red}y}} \\ \boxed{15} \\ \boxed{{\color{red}x}} \\ \boxed{14} \end{array}$

Quote:

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).

$\displaystyle \begin{array}{cccccccccccc}\text{Face up} & 1 & 3 & 5 & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \text{Face down} & 2 & 4 & {\color{red}z} & 2 & 4 & 6 & 8 & 10 & 12 & 14 & {\color{red}y} \end{array}$ .$\displaystyle \begin{array}{cccccccc}1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ 2 & 4 & 6 & 8 & 10 & 12 & 14 & {\color{red}x}\end{array}$ .$\displaystyle \begin{array}{ccccccc}1 & 3 & 5 & 7 & 9 & 11 & 13 \\ 23 & 4 & 6 & 8 & 10 & 12 & 14 \end{array}$

Quote:

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): .$\displaystyle \text{7 cards},\;{\color{red}x},\;\text{7 cards},\;{\color{red}y},\;\text{7 cards},\;{\color{red}z},\;\text{2 cards}$

Quote:

Deal it into two piles in exactly the same way.

$\displaystyle \begin{array}{cccccccccccccc}\text{Face up} & 1 & 3 & 5 & 7 & 1 & 3 & 5 & 7 & 1 & 3 & 5 & 7 & 1 \\ \text{Face down} & 2 & 4 & 6 & {\color{red}x} & 2 & 4 & 6 & {\color{red}y} & 2 & 4 & 6 & {\color{red}z} & 2\end{array}$

The face-down pile is __reversed__.

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

Quote:

Keep repeating this until you have only three cards left face down.

$\displaystyle \begin{array}{cccccccc}\text{Face up} & 1 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{Face down} & {\color{red}z} & 2 & {\color{red}y} & 2 & {\color{red}x} & 2\end{array}$

The face-down pile is __reversed__.

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

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