The Liar Paradox and the Towers of Hanoi: The 10 Greatest Math Puzzles of All Time

by Marcel Danesi

Overview

A walk through history's most mind-boggling puzzles
Ever since the Sphinx asked his legendary riddle of Oedipus, riddles, conundrums, and puzzles of all sizes have kept humankind perplexed and amused. The Liar Paradox and the Towers of Hanoi takes die-hard puzzle mavens on a tour of the world's most enduringly intriguing braintwisters, from Königsberg's Bridges and the Hanoi Towers to Fibonacci's Rabbits, the Four Color Problem, and the Magic Square. Each chapter introduces the basic puzzle, discusses the mathematics behind it, and includes exercises and answers plus additional puzzles similar to the one under discussion. Here is a veritable kaleidoscope of puzzling labyrinths, maps, bridges, and optical illusions that will keep aficionados entertained for hours.
Marcel Danesi (Etobicoke, ON, Canada) is the author of Increase Your Puzzle IQ

Product Details

ISBN-13:	9781620458136
Publisher:	TURNER PUB CO
Publication date:	08/27/2004
Pages:	258
Product dimensions:	6.00(w) x 8.90(h) x 0.80(d)

About the Author

MARCEL DANESI is the author of Increase Your Puzzle IQ (Wiley) and The Puzzle Instinct, among many other books on semiotic topics. He is Professor of Semiotics and Linguistics at the University of Toronto and Director of the Program in Semiotics and Communication Theory. Danesi has also been cross-appointed as a professor of education, having established a continuing studies mathematics program for students with difficulties in this subject.

Read an Excerpt

The Liar Paradox and the Towers of Hanoi

The Ten Greatest Math Puzzles of All Time

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By Marcel Danesi

John Wiley & Sons

ISBN: 0-471-64816-7

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Chapter One

The Riddle of the Sphinx

Let us consider that we are all partially insane. It will explain us to each other; it will unriddle many riddles; it will make clear and simple many things which are involved in haunting and harassing difficulties and obscurities now. MARK TWAIN (1835-1910)

If we visit the city of Giza in Egypt today, we cannot help but be overwhelmed by the massive sculpture known as the Great Sphinx, a creature with the head and the breasts of a woman, the body of a lion, the tail of a serpent, and the wings of a bird. Dating from before 2500 B.C., the Great Sphinx magnificently stretches 240 feet (73 meters) in length and rises about 66 feet (20 meters) above us. The width of its face measures an astounding 13 feet, 8 inches (4.17 meters).

Legend has it that a similarly enormous sphinx guarded the entrance to the ancient city of Thebes. The first recorded puzzle in human history comes out of that very legend. The Riddle of the Sphinx, as it came to be known, constitutes not only the point of departure for this book but the starting point for any study of the relationship between puzzles and mathematical ideas. As humankind's earliest puzzle, it is among the ten greatest of all time. Riddles are so common, we hardly ever reflect upon what they are. Their appeal is ageless andtimeless. When children are posed a simple riddle, such as "Why did the chicken cross the road?" without any hesitation whatsoever, they seek an answer to it, as if impelled by some unconscious mythic instinct to do so.

Readers may wonder what a riddle shrouded in mythic lore has to do with mathematics. The answer to this will become obvious as they work their way through this chapter. Simply put, in its basic structure, the Riddle of the Sphinx is a model of how so-called insight thinking unfolds. And this form of thinking undergirds all mathematical discoveries.

The Puzzle

According to legend, when Oedipus approached the city of Thebes, he encountered a gigantic sphinx guarding the entrance to the city. The menacing beast confronted the mythic hero and posed the following riddle to him, warning that if he failed to answer it correctly, he would die instantly at the sphinx's hands:

What has four feet in the morning, two at noon, and three at night?

The fearless Oedipus answered, "Humans, who crawl on all fours as babies, then walk on two legs as grown-ups, and finally need a cane in old age to get around." Upon hearing the correct answer, the astonished sphinx killed itself, and Oedipus entered Thebes as a hero for ridding the city of the terrible monster that had kept it captive for so long.

Various versions of the riddle exist. The previous one is adapted from the play Oedipus Rex by the Greek dramatist Sophocles (c. 496-406 B.C.). Following is another common statement of the riddle, also dating back to antiquity:

What is it that has one voice and yet becomes four-footed, then two-footed, and finally three-footed?

Whatever its version, the Riddle of the Sphinx is the prototype for all riddles (and puzzles, for that matter). It is intentionally constructed to harbor a nonobvious answer-namely, that life's three phases of infancy, adulthood, and old age are comparable, respectively, to the three phases of a day (morning, noon, and night). Its function in the Oedipus story, moreover, suggests that puzzles may have originated as tests of intelligence and thus as probes of human mentality. The biblical story of Samson is further proof of this. At his wedding feast, Samson, obviously wanting to impress the relatives of his wife-to-be, posed the following riddle to his Philistine guests (Judges 14:14):

Out of the eater came forth meat and out of the strong came forth sweetness.

He gave the Philistines seven days to come up with the answer, convinced that they were incapable of solving it. Samson contrived his riddle to describe something that he once witnessed-a swarm of bees that made honey in a lion's carcass. Hence, the wording of the riddle: the "eater" = "swarm of bees"; "the strong" = the "lion"; and "came forth sweetness" = "made honey." The deceitful guests, however, took advantage of the seven days to coerce the answer from Samson's wife. When they gave Samson the correct response, the mighty biblical hero became enraged and declared war against all Philistines. The ensuing conflict eventually led to his own destruction.

The ancients saw riddles as tests of intelligence and thus as a means through which they could gain knowledge. This explains why the Greek priests and priestesses (called oracles) expressed their prophecies in the form of riddles. The implicit idea was, evidently, that only people who could penetrate the language of the message would unravel its concealed prophecy.

However, not all riddles were devised to test the acumen of mythic heroes. The biblical kings Solomon and Hiram, for example, organized riddle contests simply for the pleasure of outwitting each other. The ancient Romans made riddling a recreational activity of the Saturnalia, a religious event that they celebrated from December 17 to 23. By the fourth century A.D., riddles had, in fact, become so popular for their "recreational value" that the memory of their mythic origin started to fade. In the tenth century, Arabic scholars used riddles for pedagogical reasons-namely, to train students of the law to detect linguistic ambiguities. This coincided with the establishment of the first law schools in Europe.

Shortly after the invention of the printing press in the fifteenth century, some of the first books ever printed for popular entertainment were collections of riddles. One of these, titled The Merry Book of Riddles, was published in 1575. Here is a riddle from that work:

He went to the wood and caught it, He sate him downe and sought it; Because he could not finde it, Home with him he brought it.

(answer: a thorn caught on a foot)

By the eighteenth century, riddles were regularly included in many newspapers and periodicals. Writers and scholars often composed riddles. The American inventor Benjamin Franklin (1706-1790), for instance, devised riddles under the pen name of Richard Saunders. He included them in his Poor Richard's Almanack, first published in 1732. The almanac became an unexpected success, due in large part to the popularity of its riddle section. In France, no less a literary figure than the great satirist Voltaire (1694-1778) penned brain-teasing riddles, such as the following one:

What of all things in the world is the longest, the shortest, the swiftest, the slowest, the most divisible and most extended, most regretted, most neglected, without which nothing can be done, and with which many do nothing, which destroys all that is little and ennobles all that is great?

(answer: time)

The ever-increasing popularity of riddles in the nineteenth century brought about a demand for more variety. This led to the invention of a new riddle genre, known as the charade. Charades are solved one syllable or line at a time, by unraveling the meanings suggested by separate syllables, words, or lines. In the nineteenth century, this led to the mime charade, which became, and continues to be, a highly popular game at social gatherings. It is played by members of separate teams who act out the meanings of various syllables of a word, an entire word, or a phrase in pantomime. If the answer to the charade is, for example, "baseball," the syllables base and ball of that word are the ones normally pantomimed. By the end of the century, riddles were firmly embedded in European and American recreational culture and remain so to this very day.

Mathematical Annotations

The question that the legendary sphinx asked Oedipus seems to defy an answer at first. What bizarre creature could possibly have four, then two, and finally three legs, in that order? Wresting an answer from the riddle requires us to think imaginatively, not linearly. This very type of imaginative thinking undergirds all true mathematical inquiry.

Problem-Solving

Riddles highlight how puzzles differ in general from typical mathematical problems, such as those found in school textbooks. The latter are designed to help students do something systematically (for example, add large numbers, solve equations, prove theorems, etc.). To grasp the difference, consider two typical textbook problems. Here's the first one:

Prove that the vertically opposite angles formed when two straight lines intersect are equal.

The method used to solve this type of problem is called deduction. It involves applying previous knowledge to the problem at hand.

Start by drawing a diagram that shows all the relevant features of the problem. The two straight lines can be labeled AB and CD, and two of the four vertically opposite angles formed by their intersection can be labeled x and y. One of the angles between x and y can be labeled z, as shown:

The problem asks us, in effect, to prove that x and y (being vertically opposite angles) are equal. There are, of course, two other vertically opposite angles formed by the intersection of the two lines, but they need not be considered here because the method of proof and the end result are the same. The proof hinges on previous knowledge-specifically, that a straight line is an angle of 180 degrees. Consider CD first. As a straight line, it is (as mentioned) an angle of 180 degrees. Now, notice that CD is composed of two smaller angles on the diagram, x and z. So, logically, these two must add up to 180 degrees-a statement that can be represented with the equation x + z = 180°. The equation reads as follows: "Angle x and angle z when added together equal 180°."

Now, consider AB. Notice that it, too, is composed of two smaller angles on the diagram, y and z. These two angles must also add up to 180 degrees-a fact that can be similarly represented with an equation: y + z = 180°. The two equations just discussed are listed as follows:

1. x + z = 180°

2. y + z = 180°

They can be rewritten as follows: 3. x = 180° - z 4. y = 180° - z

If you have forgotten your high school algebra, the reason we can do this is that whatever is done to one side of an equation must also be done to the other. Think of the two sides of an equation as the two pans on a balancing scale, with equal weights in each pan. The weights are analogous to the expressions on either side of an equation. If we want to maintain balance, any weight we take from one of the pans (such as the left one) we must also take from the other (the right one). In like fashion, if we subtract z from the left side of equation 1, we must also subtract it from its right side. The result is equation 3, which shows that z has been subtracted from both sides. Note that when z is subtracted from itself on the left side (z - z), it leaves 0-a result that is not normally indicated. Subtracting z from both sides of equation 2 yields equation 4.

Now, since two things that are equal to the same thing are equal to each other (for example, if Alex is six feet tall and Sarah is six feet tall, then the two people are equal in height), we can deduce that x = y, since equation 3 shows that x is equal to (180° - z), and equation 4 shows that y is equal to the same expression (180° - z). It is not necessary to figure out what the value of the expression is. Whatever it is, the fact remains that both x and y will be equal to it. We can now conclude that "any two vertically opposite angles produced by the intersection of two straight lines are equal," because we did not assign a specific value to either angle. When a proof is generalizable in this way, it is called a theorem.

Here's our second textbook problem:

Develop a formula for the number of degrees in any polygon.

Solving this problem entails a different kind of strategy, known as induction. This involves extracting a generalization on the basis of observed facts. Consider a triangle first-the polygon with the least number of sides. The sum of the angles in a triangle is 180 degrees (see chapter 5 for the relevant proof).

Next, consider any quadrilateral (a four-sided figure). ABCD is one such figure:

Notice that this figure can be divided into two triangles as shown (triangle ABC and triangle ADC). By doing this, we have discovered that the sum of the angles in the quadrilateral is equivalent to the sum of the angles in two triangles, namely, 180° + 180° = 360°.

Next, consider the case of a pentagon (a five-sided figure). ABCDE, as follows, is one such figure:

Since the pentagon can be divided into three triangles, as shown (triangle ABE, triangle BEC, and triangle ECD), we have again discovered a simple fact-namely, that the sum of its angles is equivalent to the sum of the angles in three triangles: 180° + 180° + 180° = 540°.

Continuing in this way, we can show just as easily that the number of angles in a hexagon (a six-sided figure) is equal to the sum of the angles in four triangles; in a heptagon (a seven-sided figure), to the sum of the angles in five triangles; and so on. Let's now attempt to generalize what we have apparently discovered. The letter n can be used to represent any number of sides, and the term n-gon can be used to refer to any polygon-that is, to a polygon with an unspecified number of sides. The previous observations suggest that the number of triangles that can be drawn in any polygon is "two less" than the number of sides that make up the polygon. For example, in a quadrilateral, we can draw two triangles, which is "two less" than the number of its sides (4), or (4 - 2); in a pentagon, we can draw three triangles, which is, again, "two less" than the number of its sides (5), or (5 - 2); and so on. In the case of a triangle, this rule also applies, since we can draw in it one and only one triangle (itself). This also is "two less" than the number of its sides (3), or (3 - 2). In an n-gon, therefore, we can draw (n - 2) triangles. To summarize:

Since we know that there are 180 degrees in a triangle, then there will be (4 - 2) 180° in a quadrilateral, (5 - 2) 180° in a pentagon, and so on.

Continues...

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Excerpted from The Liar Paradox and the Towers of Hanoi by Marcel Danesi Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

	Acknowledgments	vii
	Introduction	1
1	The Riddle of the Sphinx	5
2	Alcuin's River-Crossing Puzzle	27
3	Fibonacci's Rabbit Puzzle	47
4	Euler's Konigsberg Bridges Puzzle	67
5	Guthrie's Four-Color Problem	85
6	Lucas's Towers of Hanoi Puzzle	105
7	Loyd's Get Off the Earth Puzzle	125
8	Epimenides' Liar Paradox	141
9	The Lo Shu Magic Square	159
10	The Cretan Labyrinth	177
	Answers and Explanations	191
	Glossary	237
	Index	243

What People are Saying About This

From the Publisher

"As an undergraduate math major, I studied various areas of mathematics as if under a sequence of microscopes. I had no clear picture of what mathematics as a whole was about. Here is a book, based on ten great puzzles, that provides that much needed view from 40,000 feet. Big concepts such as topology, combinatorics, logic, the infinite, and many others come into clear view at this altitude. For those seeking more detail, there is always the option to land and explore a host of mathematical issues, puzzles and related references. Danesi’s approach to relating difficult material is seductive."
—Sam Savage, Consulting Professor, Stanford University and Inventor of the Shmuzzle™ Puzzle.

Editorial Reviews

"With the proliferation of puzzle books, one looks for something different, and here it is! ... This treatment will arouse interest, ally suspicion and banish fear." ("Mathematical Association of America Online")

"Delightful." ("Mathematics Teacher")

From the Publisher