Trick with 21 cards
This was the first card trick that I learned as a kid.
I deal out 21 cards faceup into three piles of seven cards each.
You silently pick one of the 21 cards while I am dealing the cards.
When I finish dealing the cards, you tell me which pile
contains the card that you chose.
I then collect the piles, placing "yours" between the other two.
We repeat this process two more times.
I then know which card is yours.
It is the 11th card in the deck, so I deal out the cards face down counting until
I get to 11, then I place that card faceup.
Note: when dealing the cards, deal one to each pile, then a second to each pile, etc.
Why does this work?
The first time the cards are dealt, the card you choose could be any 1 of the 21 cards.
When I place the pile containing your card between the other two piles, I narrow the possibilities
of which card yours could be -- 1 of the 7 middle cards in the deck (either the 8th, 9th, 10th,
11th, 12th, 13th, or 14th).
When I redeal the cards into three piles, cards 8 to 14 are either the 3rd, 4th, or 5th card in their
piles, which means that your card is either the 3rd, 4th, or 5th card in its pile.
By telling me which pile contains your card, you allow me to narrow once again the possibilities
for which card is yours. I place the pile containing your card in the middle, and then I
know that your card is either the 10th, 11th, or 12th card in the deck. This position
can be calculated by adding 3, 4 or 5 to 7 -- the first 7 cards in the deck.
When I deal again, I know your card is the 4th one in one of the three piles. When you tell me
which pile, I place it once again in the middle and am then certain that your card is the 11th one
in the deck.