Revealing someone else's magician's secret

Attempting to explain this card trick:

The Final 3 - Amazing Math Card Trick by mismag822

Ok.

So first, the three cards are chosen. That's not important.

There are 4 piles made of the remaining cards: 10, 15, 15, 9

Or, another way,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9
And your cards: 1, 2, 3

Then, you place your first card.
1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9
And your cards: 2, 3

You split the second pile, and put the second card on top of the remainder.
1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
2, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9
And your cards: 3


Then you split the third pile, and put the third card on top of the remainder.

1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 2, 3, 4, 5, 6, 7, 2, 8, 9, 10, 11, 12, 13, 14, 15
3, 8, 9, 10, 11, 12, 13, 14, 15
1, 2, 3, 4, 5, 6, 7, 8, 9


Then, you collect the cards in reverse order of the piles.
1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 2, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Notice how your splits really don't make any difference, as all 15 cards from the pile end up back together anyway. :)

So, your cards are currently 10th, 26th, and 42nd.

Then, you move the four top cards to the bottom.
5, 6, 7, 8, 9, 3, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 2, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4

Your cards are now 6th, 22nd, and 38th.

Then we do the up-down thing. As the up comes first, it will take all the odd cards, we will only be left with the even ones.

This means that your cards would be 3rd, 11th, and 19th.

HOWEVER, this also reverses the order of the cards. Therefore, the pile looks something like this:
4, 2, 10, 8, 6, 4, 2, 1, 6, 4, 2, 15, 13, 11, 9, 2, 6, 4, 2, 15, 13, 11, 9, 3, 8, 6

In other words, your cards are now 6th, 14th, and 22nd.

Even numbers again. So they'll end up downwards. Again, they would be odd (3rd, 7th, 11th), but with the cards turned around again, the cards that were even now are odd and vice-versa.

6 3, 11, 15, 4, 2, 11, 15, 4, 1, 4, 8, 2

So, in the pack of 13, your cards are 2nd, 6th, and 10th.

So, again, they'll be face-down.

Meaning that, once you turn them around again, they're 2nd, 4th, and 6th.

8, 1, 15, 2, 15 3

And once you get rid of the odd cards, well, you're left with 1, 2, 3.


How does this work? Your cards are 16 away from each other. 16 is 2 x 2 x 2 x 2. Every time the cards are whittled down, they stay an even number away. That means, if one shows up, they will all show up. Otherwise, none will. 52/16 is 3.25 (remember the odd card in the third round? that's the .25). Therefore, if you devide 52 by 2 four times, you will be left with three cards. And, if you know where to put the remainder.... you can be left with any three cards you like. ;)