497. Shuffling Cards. An elementary method of shuffling cards consists of taking a deck face down in the left hand and transferring the cards one by one to the right hand; each successive card is placed on top of the previous one: the second on top of the first, the fourth on top of the third, and so on until all the cards have been transferred. If you repeat this procedure several times with any even number of cards, you will find that after a certain number of repeated shuffles, the cards will return to their original order. If you have \(2,4,8,16,32,64\) cards, it will take \(2,3,4,5\), 6,7 shuffles respectively for the cards to return to their original order.
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* The correct order is: Ace, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Jack, Queen, King.
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How many times do you need to shuffle a deck of 14 cards?
497. To shuffle 14 cards in the manner described above and return them to their original order, it takes 14 shuffles, although in the case of 16 cards, only 5 are required. We cannot delve into the nature of this phenomenon here, but the reader may find it interesting to conduct an independent investigation of this question.

[For the mathematical theory of such shuffling, see W. W. Rouse Ball, “Mathematical Recreations and Essays” (N. Y., 1960, pp. 310-311). This shuffling is sometimes called “Monge’s shuffle,” after the famous 18th-century French mathematician Gaspard Monge, who first invented it. M. G.]