Perfect Out Shuffles-Double Down Act 4

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$7th Grade Math -- Weeks 1-2 - FCPS Curriculum, Instruction and ...$

NUMB3RS ActivityEpisode: “Double Down”The goal of this activity is to give your students a short and simple snapshot into a very extensivemath topic. TI and NCTM encourage you and your students to learn more about this topic using theextensions provided below and through your own independent research.ExtensionsFor the StudentIn this extension, you will have the opportunity to study the relationship between thenumber of cards in a deck n (where n is not a power of 2, as was the case inQuestion #4) and the number of shuffles S required to restore the deck to its originalorder.You will need to install the program SHUFFLE onto your TI-83/84 Plus calculator. Thisprogram can be downloaded for free at http://education.ti.com/exchange. Once theprogram has been loaded, run the program SHUFFLE. This program performs a perfectout-shuffle of n cards, and continues to shuffle the cards until the original order has beenrestored. The number of shuffles is displayed on the home screen at the end.Use the SHUFFLE program to complete the tables below.Numberof CardsNumber ofPerfect ShufflesNumberof Cards2 324 346 368 3810 4012 4214 4416 4618 4820 5022 5224 5426 5628 5830 60Number ofPerfect ShufflesUsing your completed tables, look for a relationship between the number of cards in thedeck and the number of perfect shuffles required to restore the deck to its original order.How many perfect out-shuffles would be needed to restore decks of 62 or 66 cards totheir original orders? Use the SHUFFLE program to check your predictions.education.ti.com/go/NUMB3RSRon Lancaster, University of Toronto, Toronto ON© 2005 Texas Instruments Incorporated
NUMB3RS ActivityEpisode: “Double Down”Shuffling and Magicians• When magicians perform a perfect out-shuffle, they do not perform it using themethod described in the activity. Skilled magicians are able to do the entire shufflewhile holding the cards in their hands, without using a table. Some magicians are ableto do this shuffle with one hand, a feat that leaves people stunned and in awe of theirskill. Details about how the deck is held and what the magician actually does can befound at http://web.superb.net/cardtric/sleights/outfaro.htm.• Another type of shuffle is called a perfect in-shuffle. To perform this shuffle, the firstcard you put on the table should be the bottom card from the pile of cards in your righthand. Using what you know about perfect out-shuffles, make a prediction about whatthe result of a perfect in-shuffle would be (think about the positions of the top andbottom cards of the original deck as well as the positions of the top card of each halfof the deck). Then check your prediction by using the 8 cards you created for theactivity to perform a perfect in-shuffle. What do you notice about the order of thecards? How might a magician use a perfect in-shuffle?• Magicians most typically use a standard deck of 52 cards. Why might a magician wantto perform a perfect out-shuffle? Think about how the magician could take advantageof the fact that the position of the top and bottom cards never change. Also, howmight a magician take advantage of the fact that the cards can be secretly restored totheir original positions?Additional ResourcesMagic Tricks, Card Shuffling and Dynamic Computer Memories, S. Brent MorrisISBN 0-88385-527-5, Mathematical Association of America, www.maa.orgThe Mathematics of Games, John D. Beasley, Oxford University Press,ISBN 0-19-286107-7Ivars Peterson’s Math Trek - Magic of Perfect Shuffles - August 1, 1998http://www.sciencenews.org/sn_arc98/8_1_98/mathland.htmShuffle: http://mathworld.wolfram.com/Shuffle.html - includes links to faro shuffle,perfect shuffle, and others.education.ti.com/go/NUMB3RSRon Lancaster, University of Toronto, Toronto ON© 2005 Texas Instruments Incorporated
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