# Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

### By Martin Gardner

Catalog Code: MGL-01
Print ISBN: 978-0-52173-525-4
208 pp., Paperbound, 2008
List Price: \$16.99
Series: Spectrum

Paradoxes and paper-folding, Moebius variations and mnemonics, fallacies, magic squares, topological curiosities, parlor tricks, and games ancient and modern, from Polyominoes, Nim, Hex, and the Tower of Hanoi to four-dimensional ticktacktoe. These mathematical recreations, clearly and cleverly presented and updated by Martin Gardner, delight and perplex while demonstrating principles of logic, probability, geometry, and other fields of mathematics.

### Table of Contents

Acknowledgments
Introduction to the First Edition
Introduction to the Second Edition
1. Hexaflexagons
2. Magic with a Matrix
3. Nine Problems
4. Ticktacktoe
5. Probability Paradoxes
6. The Icosian Game and the Tower of Hanoi
7. Curious Topological Models
8. The Game of Hex
9. Sam Loyd: America’s Greatest Puzzlist
10. Mathematical Card Tricks
11. Memorizing Numbers
12. Nine More Problems
13. Polyominoes
14. Fallacies
15. Nim and Tac Tix
16. Left or Right?
Index

### Excerpt: Ch. 10 Mathematical Card Tricks (p. 109)

Somerset Maugham's short story "Mr. Know-All" contains the following dialogue:

"Do you like card tricks?"
"No, I hate card tricks."
"Well, I'll just show you this one."

After the third trick, the victim finds an excuse to leave the room. His reaction is understandable. Most card magic is a crashing bore unless it is performed by skillfull professionals. There are, however, some "self-working" card tricks that are interesting from the mathematical standpoint.

Consider the following trick. The magician, who is seated at a table directly opposite a spectator, first reverses 20 cards anywhere in the deck. That is, he turns them face up in the pack. The spectator thoroughly shuffles the deck so that these reversed cards are randomly distributed. He then holds the deck underneath the table, where it is out of sight of everyone, and counts off 20 cards from the top. This packet of 20 cards is handed under the table to the magician.

The magician takes the packet but continues to hold it beneath the table so that he cannot see the cards. "Neither you nor I," he says, "knows how many cards are reversed in this group of 20 that you handed me. However, it is likely that the number of such cards is less than the number of reversed cards amoung the 32 that you are holding. Without looking at my cards, I am going to turn a few more face-down cards face up and attempt to bring the number of reversed cards in my packet to exaclty the same number as the number of reversed cards in yours."

The magician fumbles with his cards for a moment, pretending that he can distinguish the fronts and backs of the cards by feeling them. Then he brings the packet into view and spreads it on the table. The face-up cards are counted. Their number proves to be identical with the number of face-up cards among the 32 held by the spectator!

This remarkable trick can best be explained by reference to one of the oldest mathematical brain-teasers. Imagine that you have before you two beakers, one containing a liter of water; the other a liter of wine. One cubic centimeter of water is transferred to the beaker of wine and the wine and water mixed thoroughly. Then a cubic centimeter of the mixture is transferred to the water. Is there now more water in the wine than in the water? Or vice versa? (We ignore the fact that in practice, a mixture of water and alcohol is a trifle less than the sum of the volumes of the two liquids before they are mixed.)

The answer is that there is just as much wine in the water as water in the wine. The amusing thing about this problem is the extraordinary amount of irrelevant information involved. It is not necessary to know how much liquid there is in each beaker, how much is transferred, or how many transfers are made. It does not matter whether the mixtures are thoroughly mixed or not. It is not even essential that the two vessels hold equal amounts of liquid at the start! The only significant condition is that at the end each beaker must hold exactly as much liquid as it did at the beginning. When this obtains, then obviously if x amount of wine is missing from the wine beaker, the space previoulsy occupied by the wine must now be filled with x amount of water.

### About the Author

For 25 of his 94 years, Martin Gardner wrote “Mathematical Games and Recreations,” a monthly column for Scientific American magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of mathematics. He has also made significant contributions to magic, philosophy, children’s literature, and the debunking of pseudoscience. He has produced more than 60 books, including many best sellers, most of which are still in print. His Annotated Alice has sold more than one million copies.

Book Series: