Watch the video www.youtube.com/watch?v=C-ZfJ8OJ8n8 to see the trick revealed.
(Find instructions online for the Amazing Math Card Trick if you haven't already.)
The trick works because of how the magician arranges the cards.
If you follow the instructions, you will end up with the 3 chosen cards (from step 1) in positions 6, 22 and 38 in the deck. Why? The bottom of the deck is made of 10 cards plus the 4 you moved from the top to the bottom (step 6); the top of the deck contains the remaining 9 cards that you place there (step 4), minus the 4 cards that you moved from the top to the bottom (step 6); in between each pair of the 3 selected cards are the piles of 15. You cut the piles of 15 anywhere (step 3) and put on top the 3 chosen cards, but when you gather the separate piles into a single deck you bring the piles of 15 back together again.
Now the “proof“ — and to make it simpler on my poor typing fingers in what follows, I'm going to use “x“ to represent a chosen card and numbers to keep track of how many cards are on top, between or below chosen cards. The initial deck (after putting 4 cards top to bottom) is 5x15x15x14.
When you deal, the first 5 cards will be dealt up, down, up, down, up (ududu for short), followed by one of the chosen cards x as a down; the next 15 will be udududududududu, then another x and so on. Working through the deck this way, you will have (in the remaining deck) 2 from the first 5, x, 7 from the next 15, etc. — or 2x7x7x7. However, because you've dealt them out, you've reversed the order of the deck, so in fact you have 7x7x7x2.
Dealing again, the first 7 cards are udududu, then x, etc. to give 3x3x3x1. Again the deck is reversed, so it's really 1x3x3x3.
Next deal goes uxuduxuduxudu, giving x1x1x1, or, reversed, 1x1x1x.
Final deal goes uxuxux, leaving the 3 chosen cards face down to be turned over — magic!
In summary, the sequence of cards in the deck goes like this:
9x15x15x10 — initial stack after cutting/reassembling
5x15x15x14 — move 4 cards from top to bottom
7x7x7x2 — deal up/down, order gets reversed, 26 cards left
1x3x3x3 — deal up/down, order gets reversed, 13 cards left
1x1x1x — deal up/down, order gets reversed, 6 cards left
xxx — deal up/down, 3 cards left — the 3 chosen originally.
The above explanation is based on Misterbates’ comment on the web page www.10dailythings.com/2010/02/03/amazing-math-card-trick/.
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