Mathematical Puzzle Tales

Puzzle 1\ Lost On Capra\ \ Dr. Ziege, the eminent German extraterrestrial geologist, was the first human to set foot on Capra, the fifth planet from the star Capella. For several months she and her two companions explored the planet by spacecar. Capra is roughly twice the size of the earth, but lacking in enough water to support life. Dr. Ziege found the planet a barren, sandy waste, its surface as smooth as the plains of Kansas. Like the earth, Capra rotates on an axis. Dr. Ziege designated one pole north and the other south in conformity with the ship's magnetic compass and the planet's earthlike magnetic field. Geographic and magnetic Caprian poles coincide. The last radio message from Dr. Ziege was: "we have lost our bearings and cannot find the spaceship. Yesterday we drove 10 myriamaters due south from our last camp site, then 10 myriameters due east, then 10 myriameters due north. We find ourselves back at the camp site. Food supplies exhausted. Send help. Attempts to reach Dr. Ziege for precise information as to her location brought no response. The German government immediately fired a rescue ship through Wheeler wormhold 124C41+. Two days later it was circling Capra with plans to land near the north pole. It seemed obvious that only from that pole could Dr. Ziege and her men go 10 myriameters south, then east, then north and be back at the starting spot. But there were no signs of the explorers within a radius of 20 myriameters from the north pole. "Ach!" shouted Felix, striking his temples. "We are looking in the wrong place. Another spot fits Dr. Ziege's message perfectly.""How can that be?" said Hilda. "If the starting spot is a few kilometers from the north pole, the terminal spot will miss the pole by a short distance. The farther south you go, the more it misses. At the equator it misses the starting spot by a full ten myriameters. And south of the equator it will miss by more than that!"Nevertheless Felix was right. Where should they look next? Puzzle 2 The Doctors' Dilemma\ The first earth colony on Mars has been swept by an epidemic of Barsoomian flu. The cause: a native Martian virus not yet isolated.\ There is no way to identify a newly infected person until the symptoms appear weeks later. The flu is highly contagious, but only by direct contact. The virus transfers readily from flesh to flesh, or from flesh to any object which in turn can contaminate any flesh it touches. Residents are going to extreme lengths to avoid touching one another, or touching objects that may be contaminated.\ Ms. Hooker, director of the colony, has been seriously injured in a rocket accident. Three immediate operations are required. The first will be performed by Dr. Xenophoon, the second by Dr. Ypsilanti, the third by Dr. Zeno. Any of the surgeons may be infected with Barsoomian flu. Ms. Hooker, too, may have caught the disease. Just before the first operation it is discovered that the colony's hospital has only two pairs of sterile surgeon's gloves. No others are obtainable and there is no time for resterilizing. Each surgeon must operate with both hands. "I don't see how we can avoid the risk of one of us becoming infected," says Dr. Xenophon to Dr. Zeno. "When I operate, my hands may contaminate the insides of my gloves. Ms. Hooker's body may contaminate the outsides. The same thing will happen to the gloves worn by Dr. Ypsilanti. When it's your turn, you'll have to wear gloves that could be contaminated on both sides.""Au contraire," says Dr. Zeno, who had taken a course in topology when he was a young medical student in Paris. "There's a simple procedure that will eliminate all risk of any of us catching the flue from one another or from Ms. Hooker."\ What does Dr. Zeno have in mind? Puzzle 3: Space Tool\ Two young physicists were discussing their vacation plans. \ "I may take a space cruise," said Jones. "I've been told that the food and the girls on the Cutty Snark are superb, and that this summer the cruise includes landings on the moon, Mars, and Venus.""I went last year," said Smith, "and had a marvelous time. The ship has a huge recreation room with all sorts of new games. Spacec pool, for instance. When the ship's in a g-field it's played the regular way, only the table is enormous and there are more than 100 balls." "How is it played in zero gravity?""Some engineer figured out a way to create a green-tinted magnetic force field," said Smith. "It keeps the balls inside a rectangular parllelepiped about a meter above the table. The ivory balls have iron cores. They bounce off the green walls the same way they bounce off the cushions on the table. The wooden cues are not affected by the field, so you can poke them into the field at any spot. The pockets are holes in the field's eight corners. If a ball hits a corner it leaves the field and you score the ball's number like in ordinary pool.""Bust won't the balls keep on moving after they're hit? How can you storke the cue ball when it's on the wing?""The balls freeze exactly ten seconds after each stroke," said Smith. "I don't know how it works. I think another force field brings all the balls to a dead stop."\ "How many balls are there?"\ "I can't recall. Somewhere between one and two hundred. When the game's played on the table it starts with the balls packed into a triangle like the 15 balls of regular pool. When it's played in space you start with the same set of balls packed into a regular tetrahedron.""In other words," said Jones, "the number of balls is both triangular and tetrahedral. There can't be many numbers like that."Smith closed his eyes. "Well, there's 1. It's triangular and tetrahedral, but that's trivial. The next tetrahedron is a triangle of 3 balls with 1 ball on top, or 4 altogether. But 4 balls won't make a triangle.""Ten will," said Jones. "It makes a triangle with rows of 1, 2, 3, and 4. And it also makes a tetrahedron. Every tetrahedral number is the sum of consecutive triangles; and triangles 1, 3, and 6 add to 10."Jones took out his calculator. "Let's see. If I remember my number theory, triangular numbers have the form ½ n(n + 1) where n is any positive integer. Tetrahedral numbers have the form 1/6 n(n + 1) (n + 2)."It didn't take Jones long to discover that the third number to fit both formulas was between 100 and 200. He could find no other solution less than 200, so this was the number he wanted. With the aid of a pocket calculator, how quickly can you determine the number of balls (not counting the cue ball) used in space pool?Puzzle 4: Machismo on Byronia\ Byronia, a small planet that orbits a sun near ours, has a humanoid population similar to our own. The most striking difference is that Byronians come in three sexes. They correspond roughly to what we call male, female, and bisexual.Because bisexuals have both male and female organs, they can perform as either sex and also bear children. Whenever a "mother" (female or bisexual) gives birth, the probability that the child is male, female or bisexual is exactly one-third for each. The new Supreme Ruler of Byronia, Norman Machismo, is a virile, hot-tempered male who gained total power by defeating a rebellious army of bisexuals. To solve the "bisexual problem" Machismo has issued the following decree: Every mother on Byronia, as soon as she or it gives birth to a bisexual, is to be rendered incapable of further conception. Machismo reasoned like this. Some mothers are sure to have two, three, four, or even more heterosexuals before having a bisexual. True, occasionally a mother will have a bisexual first child, but that will be the end of her childbearing, so these births will contribute only a small percentage of bisexuals to the population. In this way the proportion of bisexuals in the populations will steadily diminish.Will the Supreme Ruler's plan work?

Foreword  vii Preface xi  How to Use This Book  xii Puzzles 1  Lost on Capra  3 2  The Doctors' Dilemma  5 3  Space Pool  6 4  Machismo on Byronia  8 5  The Defective Doyles  9 6  The Third Dr. Moreau  10 7  The Voyage of the Bagel  11 8  The Great Ring of Neptune  12 9  The Toroids of Dr. Klonefake  13 10  The Postage Stamps of Philo Tate  15 11  Captain Tittlebaum's Test  17 12  Exploring Carter's Crater  20 13  Pink, Blue, and Green  22 14  The Three Robots of Professor Tinker  23 15  How Bagson Bagged a Board Game  24 16  The Shop on Bedford Street  26 17  Tanya Tackles Topology  28 18  The Explosion of Blabbage's Oracle  30 19  Dracula Makes a Martini  32 20  The Erasing of Philbert the Fudger  33 21  Oulipo Wordplay  35 22  How Crock and Witson Cracked a Code  37 23  Titan's Titanic Symbol  38 24  Professor Cracker's Antitelephone  40 25  Vacation on the Moon  41 26  Weird Numbers From Titan  43 27  Lucifer at Las Vegas  45 28  Off We're Going to Shuttle  46 29  The Backward Banana  48 30  The Queer Story of Gardner's Magazine  50 31  Blabbage's Decision Paradox  53 32  No Vacancy at Aleph-Null Inn  55 33  Tube Through the Earth  57 34  Robots of Oz  58 35  The Dance of the Jolly Green Digits  61 36  The Bagel Heads Home  63 Answers First Answers  67 Second Answers  103 Third Answers  129 Bibliography  147 Postscript  149

\ The Mathematical Gazette"Those new to this field of recreational problems will find much here to amuse and interest them, and if their appetite is sufficiently whtted they will find very useful a bibliography of 35 relevant items. It is certainly worthy of a place on the shelves of the school library."\ \ \ \ \ Zentralblatt fur Mathematik"As is often the case with his books, Martin Gardner generates delightful trains of thought. A solution to a riddle appears in "First Answers," where it generates new questions, which in turn are solved in "Second Answers," where they give birth to a new generation of questions answered in the "Third Answers." In this new edition Martin added a three-page postscript that summarizes new information related to some of the riddles….The book is well written. Every curious person would enjoy reading these essays and pondering on problems they raise, for whatever interests Gardner, he makes interesting for everyone! Thus, the book would be a worthy educational entertainment for the broadest audience of those who enjoy engaging in gymnastics of brain, which keeps the latter flexible."\ \ \ BooknewsGardner, author of the Mathematical Games column for for 25 years, wrote the 36 stories collected here for