How many shuffles?
The rudimentary method of shuffling a pack of cards is to take the pack face downwards in the left hand and then transfer them one by one to the right hand, putting the second on top of the first, the third under, the fourth above, and so on until all are transferred. If you do this with any even number of cards and keep on repeating the shuffle in the same way, the cards will in due time return to their original order.
Try with 4 cards, and you will find the order is restored in three shuffles. In fact, where the number of cards is 2, 4, 8, 16, 32, 64, the number of shuffles required to get them back to the original arrangement is 2, 3, 4, 5, 6, 7 respectively. How many shuffles are necessary in the case of 14 cards?
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To shuffle fourteen cards in the manner described, so that the cards shall return to their original order, requires fourteen shuffles, though with sixteen cards we require only 5.
The engineering department was arranging for a rather expensive catered lunch to bid farewell to their retiring colleague. They calculated that it would cost each person $30. One good mathematician remarked, "It's lucky that there aren't five fewer of us to split the...