A magician asks a person to select a card from a pack of 27 cards and return it back without showing it back to the magician. After the choice is made, the magician shuffles the pack and deals all the cards face up into three piles, one card to each pile in turn. The person is now asked which pile contains the card. The pile is then placed between the other two piles and without any shuffling, the cards are again distributed again into three piles. The same procedure is repeated again and for the final time the cards are distributed in three piles and the person is asked about the pile containing the card. The magician then names the selected card. Explain the trick.
Let us denote the cards in the three piles after the first deal as, say, a1, a2, …a9; b1, b2, …b9; c1, c2, …c9. If the selected card is, to be specific in pile 1 after the second deal the piles will look as shown in layout 2. Note that all the cards that were in pile 1 after the first deal are now exactly in the three middle positions in each of the piles. If the selected card is now in, say, pile 3, that is, either a3, a6, or a9, it will be exactly in the middle of a pile in the final distribution. Hence, pointing out the pile containing the selected card in the final distribution uniquely identifies that card. So, the selected card always lies at the center of the pile containing it in the final distribution.