A performer takes 21 cards from a standard deck and shuffles them. A player notes one at random. The performer deals out the cards into three columns of seven cards each. The player indicates the column that contains her card. Twice more the performer deals out the cards into three columns and the player identifies the one containing her card. At this point the performer identifies the card.
How is this done? The trick works automatically so long as, in taking up the cards, the performer always puts the chosen pile between the other two. After the first deal, the chosen card will fall in one of positions 8-14; after the second deal, it will reach position 10-12; and after the last deal, it will be the 11th card in the assembled packet (at which point the performer can reveal it however he pleases).
This illustration, by CMG Lee, demonstrates the same principle using 27 cards. At each step, the pile containing the chosen card is shaded yellow; the numbers correspond to the step numbers. In this case the chosen card always finds its way to the 14th position.
Here’s an animation with 21 cards, in which the chosen card, marked with an X, finds its way to the 11th position:
