This essay evolved from reading the chapter on George Sands' "Prime Number Principle" in Arthur F. MacTier's new book: Card Concepts (Lewis Davenport, 2000). For those who don't know, here's Sands' effect in brief: begin with a prime number of cards P = 3, 5, 7, 11, or 13,..Note the top card. Pick any number N less than the number of cards in the packet. Transfer N cards from top to bottom, then turn over the top card. Do this P-1 times in total. The card noted will be the only card remaining face-down in the packet.
When we transfer cards from the top to the bottom of a packet of cards, in effect we're creating a string of cards that can be as long as we like. If there is a prime number of cards in the packet—let's say seven—and we transfer any number N less than seven (more on this later), then we are counting: N-2N-3N-4N-5N-6N. Because seven is a prime number—that is, it has no divisors other than one and itself—none of those numbers can possibly be a multiple of seven. Hence none of those numbers can ever turn up either the original top card or any of the numbers previously on top. Finally we come to 7N, which is a multiple of seven, so we are back to the original top card.
That takes a little reflection to get one's head around, but thinking of it that way leads to an extension of the Prime Number Principle, which we'll call the Co-Prime Number Principle:
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