4 Cards are being removed from the deck (always from top of 9 pile)

so the piles are now: 10 | 15 | 15 | 5

since you pick up the deck in reverse order, the stack (from top to bottom) will look like:

5 card1 15 card2 15 card3 10 –48 cards

48 = 3*(2^4), you throw out half the cards 4 times (24 12 6 3) to remain with 3.

However, when you deal the cards face down (the “remaining” pile) you reverse the order. Therefore the first round removes all the odd cards (1,3,5…) the second round removes every other card starting from the last (48,44,40…), the third is every other from the start again (2,10,18…) and reverse for the fourth.

This leaves cards 6,22,38… so the setup is just a fancy way of arranging the cards for the “trick”.

here is what happens:

The whole cutting the piles is a ruse. When you put the ace on the first pile there are 10 cards and then an ace. You cut the 15 into two parts x and y. You place x on top of the ace and then put the second ace on top of the y. You cut the last 15 pile into two parts c and d. you put c on top of the “ace y” pile and the ace on top of the d. the remaining 9 cards go on top of the “ace d”.

The piles are as such at this point from left to right and listed top to bottom:
9 ace d
c ace y
x ace 10

assembly creates the following pile from top to bottom:
9 ace d c ace y x ace 10

since d c = 15 and x y = 15:
9 ace 15 ace 15 ace 10

move 4 from the top to the bottom:
5 ace 15 ace 15 ace 14

Then it getting annoying to track. The first pass of “up then down” removes every other card while reversing the stack. the stack goes from “5 ace 15 ace 15 ace 14″ to “7 ace 7 ace 7 ace 2″. The next pass reduces it to “1 ace 3 ace 3 ace 3″. Another pass reduces it to “1 ace 1 ace 1 ace”.

Your deck will ALWAYS look like this after the re-stacking (from the top):

9 cards – third marked crad – 15 cards – second marked card – 15 cards – first marked card – 10 cards

The spacing is chosen so that doing the repeated ‘up-down’ split will leave te three markled cards for last.