The Final 3 Card Trick Explained

The first part to this magic trick uses what I think is called a "false cut" in magic.

Let's refer to the stacks as follows:

Stack of 10 = Stack 1
First stack of 15 = Stack 2
Second stack of 15 = Stack 3
Stack of 9 = Stack 4

The subject is told to put the first card, let's call it card 1, on top of stack 1.

Then there is a cut. Let's say the subject puts x cards on of card 1.

Then the subject is told to put card 2 on top of stack 2.

Then the subject is told to cut the third stack on top of card 2. Let's say he cuts y cards on top of card 2.

Then the subject it told to put card 3 on top of stack 3.

The the subject puts stack 4 on top of card 3. So there will be 9 cards above card 3.

Then the subject puts the that stack on top of stack 2. There must be 15-y cards under card 3. After placing that stack on top of stack 2 there must be 15-y + y = 15 cards between cards 2 and 3. So, the subject's cards are now the 10th and 26th cards from the top of the left stack.

The the subject moves the left stack on top of stack 1. There must be 15-x cards between card 2 and the bottom, and x cards on top of card 1. After moving the stack, there will bet 15-x+x=15 cards between cards 1 and 2. So, the subject's cards are now the 10th, 26th, and 42nd cards from the top of the left stack.

Then the magician moves the top four cards to the bottom. Then the subject's cards will be in positions 6, 22, and 38 from the top.

In the next step the magician gets rid of every odd numbered card. Doing that alone would change the original cards to positions 3, 11, and 19. However, in the process of flipping them over, the magician reverses the order. With 26 cards left, the original cards are now in positions 8, 16, and 24.

The magician goes through put the cards in alternating piles, effectively removing the odd numbered cards. That alone would move the original cards to positions 4, 8, and 12. However, again, he reverses the order. There are 26 cards left at this point, so the original cards are now in position 2, 6, and 10.

In the next step the magician gets rid of every odd numbered card. Doing that alone would change the original cards to positions 1, 3, and 5. However, in the process of flipping them over, the magician reverses the order. With 13 cards left, the original cards are now in positions 2, 4, and 6.

The magician goes through put the cards in alternating piles, effectively removing the odd numbered cards. That alone would move the original cards to positions 1, 3, and 5. However, again, he reverses the order. There are 6 cards left at this point, so the original cards are now in position 2, 4, and 6.

With one more alternation the original cards are in positions 1, 2, and 3. Ta da!