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Can you explain the card trick that is explained here?

Edit: Here's a summary of the trick as explained in the video:

Start by asking a spectator to pick any three cards they like out of a standard 52-card deck, without showing them to you, and write them down (to make sure they won't forget them). (Ed. note: You can shuffle the deck if you like, or even let the spectator shuffle it, but you don't have to.)

Divide the remaining cards into four piles, so that first pile will have 10 cards, the second and third piles will have 15 cards each, and fourth pile, set aside, will have the remaining 9 cards.

Now, tell the spectator to put the first card they picked on top of the first pile, then cut the second pile anywhere they want and put the top half on top of the first pile (and the card they picked). Then they should put the second card they picked on top of what remains of the second pile, cut the third pile anywhere they want and put the top half on top of the second pile, and finally put the third card they picked on what remains of the third pile and place the entire fourth pile on top of it. (Ed. note: You could have the spectator cut the fourth pile too, if you wanted; it shouldn't matter as long as all nine cards of it eventually end up on top of the third pile.)

Now collect the three piles of cards together so that pile #3 ends up on top of the deck, pile #2 in between and pile #1 on the bottom. Next, take four cards off the top of the deck and place them on the bottom. Deal the cards from the top of the deck alternatingly into two piles, the first pile face up and the second pile face down. Tell the spectator in advance to say "stop" if they see any of their cards in the face-up pile (which they won't).

Once you've dealt out the entire deck, set the face-up pile aside, pick up the face-down pile and repeat the process, dealing it into two smaller piles, the first pile face up and the other face down. Again, tell the spectator to say "stop" if they see any of their three cards in the face-up pile — they won't. Keep repeating this process until you're down to just three face-down cards. Show those cards to the spectator; they'll be exactly the ones they picked and wrote down.

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    I disagree with closing this question. It would be much better to make the question self-contained, by describing the trick in detail, so people didn't have to follow the link. But it is a real question with a mathematical answer, as Marvis has shown.2012-12-22
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    It was such an involved trick that I thought the video would be the most efficient way to explain all that is going on.2012-12-22
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    I would suggest that typing up the procedure would certainly prevent link rot, and might have made you think about $15=2^4-1$ as an important clue to the explanation. The repeated dealing and eliminating every other card cries out for an explanation based in powers of two.2012-12-22
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    If you want to type it out, have at it. Otherwise, close it.2012-12-22
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    I was primarily wanting to prevent closure. But the trick isn't too complicated, and I find explaining a problem to someone a good way to focus my thinking. The face up/face down deal to remove alternate cards is a standard technique. Recognizing that there are only two kinds of cards-those picked by the spectator and the others can help understanding. Then you can see that the cuts don't matter.2012-12-22
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    stan's latest comment made me quite sure voting to close is the right move.2012-12-22
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    Come on! This question was asked only 8 hours ago, but the external content is not accessible. This is really bad! Present the card trick here..2012-12-22
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    Link rot within one day. Classic.2012-12-22
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    This question (1) doesn't deserve to be closed as it is a mathematical one, and (2) is of the rather few instances I can think of that a link is way better than a self contained explanation, which would have made the question long and cumbersome.2012-12-22
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    Link still working for me.2012-12-22
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    @Ilmari Great Job!2012-12-22
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    @Ilmari This phrase, "shuffle it, but it might be better not to, since not " is a bit confusing. I guess you wanted to say "it might be **better to**"2012-12-22
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    @Marvis: I meant "not to", but I can see how it could be confusing. It's a bit long for an editorial note anyway, so let me simplify it.2012-12-22

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