09-4-2008, 04:05 PM | #1 |
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[High School - Calculus/Algebra/General Math] Card Shuffling Problem
Today our Calc teacher gave us a practice quiz for the upcoming year. On the quiz were a couple of trick problems that seemed hard but once you found out how to do them, they were much simpler. One of the problems, however, has given me a headache because I have been thinking throughout the day what the answer could be.
The question: if you shuffle a deck of cards perfectly starting with the left hand on top (meaning the first shuffle will be A,A,2,2,3,3,4,4 and so on), how many times will you have to shuffle it to return to its original position? I did a few perfect shuffles, and I kept track of the 2nd card, the 2 of Spades. After a while, I noticed that the distance it travels throughout the deck keeps doubling - after 0 shuffles, its the 2nd card, after 1 shuffle its the 3rd card, after 2 shuffles, its the 5th card, and so on. Now comes the tricky part. Because my numbers go past 52 and keep getting larger, its obviously not going to come back to the 2nd position. Will I know that the card has reached its original position when its place in the deck is a multiple of 52 plus 2 because its the 2nd card, or is it plus another number? Or, is my math totally wrong and I need to start from scratch? Please, any help (without giving me the answer, I want to find it myself for future problems) would be much appreciated. |
09-4-2008, 04:12 PM | #2 |
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Re: [High School - Calculus/Algebra/General Math] Card Shuffling Problem
There is no such thing as a "perfect shuffle."
Unless your teacher is dumb, it's a trick question. There is no telling which card would fall on bottom first, so on. EDIT: Unless you left out vital details, it's kind of impossible for us to answer.
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09-4-2008, 04:25 PM | #3 |
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Re: [High School - Calculus/Algebra/General Math] Card Shuffling Problem
you should start with a smaller number and look for a pattern, then scale it. at least, to me, that's the easiest method. i'm 100% sure there is a mathematical solution as well that would give the same result.
Like.. assume that you are shuffling chips at a poker table, instead of cards. Same deal though. if you have 6 chips, how many shuffles will it take to get the chips in the same order? If you have a stack with 123456, you'll have: 1 shuffle: 142536 2 shuffles: 154326 3 shuffles: 135246 4 shuffles: 123456 now let's try with 4: 1234 1st shuffle: 1324 2nd shuffle: 1234 let's try with 8: 12345678 1st shuffle: 15263748 2nd shuffle: 13572468 3rd shuffle: 12345678 now 10: 123456789A 1st: 162738495A 2nd: 186429753A 3rd: 198765432A 4th: 159483726A 5th: 135792468A 6th: 123456789A 4:2, 6:4, 8:3, 10:6... i'm sure there is a pattern in there somewhere. maybe you need to do it further out or something though. edit: i bet it has to do with odds/evens and how many are in each stack. you have 2 in each stack, it takes 2 times. you have 4 in each stack, it takes 3 times. you have 3 in each stack, it takes 4 times. you have 5 in each stack, it takes 6 times. let's try with 6 in each stack (aka, 12)... 123456789ABC 1st: 1728394A5B6C 2nd: 147A258B369C 3rd: 184B73A6295C 4th: 1A8642B9753C 5th: 1BA98765432C 6th: 16B5A493827C 7th: 1963B852A74C 8th: 15926A37B48C 9th: 13579B2468AC 10th: 123456789ABC ok... so 2:2, 3:4, 4:3, 5:6, 6:10. makes my theory not right. you're trying to find 26. meh. oh well. probably something dealing with powers of 2.
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RIP Last edited by Tasselfoot; 09-4-2008 at 04:39 PM.. |
09-4-2008, 04:51 PM | #4 |
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Re: [High School - Calculus/Algebra/General Math] Card Shuffling Problem
@Mr. Nothing: What I mean by a perfect shuffle is that the deck is split 26 each, and then each side is layered, meaning the first shuffle of an unshuffled deck would be A,A,2,2,3,3,4,4,5,5 and so on.
@Tass: I tried that, but after 12 it starts to become tedious. Would my method work at all? I ask this because once we find the correct answer, we have to explain our methods, and it would be easier to explain my own method than someone else's. |
09-4-2008, 04:58 PM | #5 | |
Retired BOSS
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Re: [High School - Calculus/Algebra/General Math] Card Shuffling Problem
just did a little research... the answer is 8. although that only works for a 52 card deck. if you play pinochle, which is 48 cards, it'll be different.
if you want to write it down on paper and try it out... it should only take 8 shuffles. edit: Quote:
edit2: for a 48 card deck, you'd have to shuffle 23 times. 2^23 = 8388608, and 47 goes evenly into 8388607.
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RIP Last edited by Tasselfoot; 09-4-2008 at 05:07 PM.. |
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09-18-2008, 03:39 PM | #6 |
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Re: [High School - Calculus/Algebra/General Math] Card Shuffling Problem
Wow! Good job tassle!
This stuff makes my brain hurt... (guessing I won't be counting cards anytime soon)
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