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#11
08-04-2005, 08:34 PM
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Re: math problem
[ QUOTE ]
I was at Foxwoods the other night, and instead of trying to have winning session, I was concentrating on chip shuffling. I was trying to figure out a formula for how many iterations it takes two stacks of n chips to get back to their original configuration, assuming you shuffle them perfectly, and the stack on the left always contributes the bottom chip to the shuffled stack. [/ QUOTE ] is the stack on the left the top half or the bottom half of the single stack? 
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#12
08-04-2005, 10:58 PM
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Re: math problem
[ QUOTE ]
[ QUOTE ] I was at Foxwoods the other night, and instead of trying to have winning session, I was concentrating on chip shuffling. I was trying to figure out a formula for how many iterations it takes two stacks of n chips to get back to their original configuration, assuming you shuffle them perfectly, and the stack on the left always contributes the bottom chip to the shuffled stack. [/ QUOTE ] is the stack on the left the top half or the bottom half of the single stack? [/ QUOTE ] The top half of the single stack always becomes the right stack.
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#13
08-06-2005, 10:13 PM
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Re: math problem
[ QUOTE ]
[ QUOTE ] [ QUOTE ] I was at Foxwoods the other night, and instead of trying to have winning session, I was concentrating on chip shuffling. I was trying to figure out a formula for how many iterations it takes two stacks of n chips to get back to their original configuration, assuming you shuffle them perfectly, and the stack on the left always contributes the bottom chip to the shuffled stack. [/ QUOTE ] is the stack on the left the top half or the bottom half of the single stack? [/ QUOTE ] The top half of the single stack always becomes the right stack. [/ QUOTE ] this means the bottom chip never changes. dumb.
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#15
08-07-2005, 03:57 AM
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Re: math problem
[ QUOTE ]
If you express the shuffle as a permutation in cycle notation the least common multiple of the cycle lengths is the number of shuffles required. This is basically restating the period thing earlier, but provides a more systematic approach. Now I simply need a quick way to get the cycle lengths and I will be done. [/ QUOTE ] yeah, we figured out that number of shuffles required is the LCM of all of the cycle lengths. one problem i thought of with using permutation groups was that if you number the chips, say, 1 2 3 4 5 6 7 8, with 1 2 3 4 being the left stack from bottom to top (and the two stacks are different colors, obviously), then the permutation group answer will tell you how many riffles you need to return to 1 2 3 4 5 6 7 8. but there are other visually identical permutations like 1 3 2 4 5 7 6 8 that maybe occur before the original order occurs again, which would mean that the number of riffles needed is less than what the permutation answer would say. hope that isn't too convoluted.
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#16
08-07-2005, 08:48 AM
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Re: math problem
Here is a link I found.
Shuffle stuff [ QUOTE ] It turns out that an ordinary deck of 52 cards is returned to its original order after eight out-shuffles. It's also possible to move the top card of a deck to any location by using the right combination of in- and out-shuffles. [/ QUOTE ] There is an interesting related issue. How many shuffles does it take to randomize deck? Any one care to research? I remember reading about this back in my blackjack days, when I was keen on shuffle tracking -- though I never got anywhere with it in practice.
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