Take two numbers M and N that are co-primes; that is, they have no factors in common. If we have a packet of M cards, transfer N cards from the top to the bottom each time, then turn over the top card, then we will turn over a new card each time, and the original top card will only be turned over after the M-th transfer.
For example, if we have a nine card packet and we transfer four cards each time, the Sands Principle will work, even though neither nine nor four is prime. But they are co-prime: 9 = 3 X 3 and 4 = 2 X 2. Hence they have no factors in common.
Try a counter-example where two numbers are not co-prime. Let's say we have a six card packet and transfer four at a time. Both four and six have a factor of two in common, so we know it won't work. Let's try it: 4-8-12. 12 is not only 4 X 3; it's also 6 X 2, so we're back to the top card.
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