however imperfectly

*Perfect* shuffles result in *relatively* easily predicted orderings. Imagine a deck of 52 cards, numbered from 1 to 52 and in ascending order. Perfectly shuffle them, and they are ordered 1, 27, 2, 28, … or 27, 1, 28, 2, …. Of course, the extreme of imperfect shuffling (perfectly imperfect, as it were) basically just cuts the deck only to slap it back together exactly as it was.

Obviously, if an observer knows the prior order of the deck, and the shuffling proceeds in some manner that allows the observer to note and record how many cards are flipped from each side before more come from the other, then the new order could be completely predicted. This would be true no matter *how* many times the deck were shuffled. So when shuffling works, it is by some combination of speed and of concealment on the part of the shuffler, and of inattention (driven by honor, by sloth, or by limitations of ability) on the part of observers. Any mathematical

proof is going to entail some assumptions (implicit or otherwise) about that to which the observer attends.

To say that something is without trend is to say that it is subject to a description that can be decomposed into a hyperplane on the axes of the explanatory variables (typically just a time dimension), and deviations from that hyperplane that are simply *not explained* by the model. These deviations may not be worth the *bother* of explanation, but that doesn't mean that they are not chaotic or not disorderly or not *anything* except that they are not *explained*.

http://www.dartmouth.edu/~chance/course/topics/winning_number.html

Specifically, I talk about the bridge players in New York who were found to be taking advantage of the fact that they decks were not being shuffled more than three times at their club:

Dr. Diaconis has found that many bridge players take advantage of the non-randomness of seemingly shuffled cards. He said a bridge club in New York State once consulted him, as a magician, to find out whether several players were cheating. After watching play ''and doing a little thinking in between,'' Dr. Diaconis knew what was going on. These players had figured out that the cards were not being randomly shuffled, and that they could predict the

distributions of cards by knowing what the deck looked like at the end of the previous hand.The players ''admitted to it readily,'' Dr. Diaconis said. ''But they didn't think they were doing anything wrong. After all, they were just thinking.'' The club asked those players not to play together for a year.

I steadfastly shuffle seven times whenever the cards are going to played in a competition. For practice rounds, admittedly, I do not always do the full seven shuffles. Interestingly, there are still bridge players who insist that the computer dealt hands are "fixed", but I think what is actually occurring is that they are accustomed to people shuffling the cards three times and thinking, "Close enough."

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