What's going on with the math?
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Take another look at the example of a perfect shuffle based on 20 cards from the
what is a perfect shuffle page. Choose a card and follow its path through each shuffle. For example, 1 starts of course as the first card. It then moves to the 2nd spot, 4th spot, 8th spot, 16th spot. See what is happening? The number doubles with each perfect shuffle. By this rule the card should then move to the 32nd spot. But there are only 20 cards and it instead moves to the 11th spot. Why the 11th spot? Because the numbers are actually doubling in modulo 21. 32 ≡ 11 (mod 21). 32 is equivalent to 11 mod 21. Completely baffled by that statement? See my explanation of what is modular arithmetic.
If you choose another card to follow you will see that again the spot it is in after
each perfect shuffle doubles in mod 21. |
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In the example of 20 cards we have examined it took 6 perfect shuffles for the cards
to return to their original order. It takes 5 colors to color the pattern.
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| The patterns created by all perfect shuffles are not so beautiful. Depending on the number of cards the pattern may include many different colored groups or it may be that the cards rotate around to every place before returning to their original configuration leaving only one color group. 12 cards is an example of a shuffle that that does this. |
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| I chose to knit scarves based on prefect shuffles of 76, 114, & 142 cards because these numbers each produce a pattern with 4 or 5 colors. When designing scarves it becomes useful to be able to predict the number and size of the color groups that will be created by a certain size of perfect shuffle. It turns out the color groups depend on the factors of the number one larger than the number of cards, the number used for the modulo. |
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