## Penelopes Principle

When, in 1957. Mr. Elmsley published his series of articles on faro shuffle principles and their mathematics, in his closing lines he mentioned having reserved one principle in particular for his private use. This was obviously a tool that he valued highly. Over the years it was passed quietly from hand to hand through the inner circles of cardinen, and as was inevitable, tricks based on this ingenious principle began to appear in print—sometimes with credit given to its inventor, but more often not. Mr. Elmsley did not formally release Penelope's principle, for that was its name, until 1988, over thirty years after its formulation. Penelope was the daughter of Icarius and the fabulously faithful wife of Odysseus, who, during Odysseus' twenty-year absence, wove and at night unwove a tapestry, at the completion of which she had vowed to make a choice from importuning suitors. Mr. Elmsley's unending tapestry is the woven deck.

The principle is this: Assume you have a particular card—say the ace of spades—at a position twenty-sixth from the top of the pack. If a spectator then cuts a small packet from the bottom of the deck and you follow this with a perfect out-faro of the remaining cards, the ace of spades will now be at a position from the top equal to the number of cards cut away by the spectator.

Some further explanation ts in order. The faro weave must be star ted from the bottoms of the packets for the principle to work consistently. Those who weave from the top down will find that, when an odd number of cards is cut away, the target card will not be positioned by the shuffle as desired. But if the weave is started at the bottoms of the packets, the principle is entirely dependable. This holds whether the upper portion contains one less card than the lower, or one more. (In the latter case the weave will end with the top two cards of the upper portion left unwoven.)

Those who weave downward may wish to turn the above process topsy-turvy. The target card in this case is located twenty-seventh from the top. The spectator is asked to cut a small packet from the top of the deck. If you now perform an out-faro with the remainder, beginning the weave from the top, the target card will lie as many cards from the face of the pack as are contained in the removed packet.

Mathematically, the principle can be expressed in this way: If a card rests within the mid-portion of the deck, and if "x" cards above it and "y" cards below it are removed, and if the remainder Is given a faro shuffle, the card of concern will be transported to a position k + y - x from the top, in which "k" is some constant that depends on the original position of the card and on the type of weave.

The following four tricks are illustrations of how Penelope's principle is put to use.

August 1988

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