Author: Ian Stewart
Year Published: 2010
Review: Cows in the Maze is the third collection of Ian Stewart's contributions to the Mathematical Explorations column of <i>Scientific American.</i> This is a book of recreational math, and the requisite level of math is, for the most part, quite low.. That is, you do not need anything more than high school math - for most of the columns, grade school math will be enough. But you do need mathematical curiosity. If you have even a little of this, then this book will likely give you more. Even though the mathematical level is low in terms of formal requirements, the mathematical sophistication level is high - there are no "If a train leaves Boston at 5 AM and another train leaves NY at 6 AM" type of problems here. This is math.
Cows in the Maze is divided into 21 chapters, most of which can be read independently and in any order. Each is about 10 to 15 pages long. They are:
1. The lore and lure of dice - about some aspects of the probability of dice (see below)
2. Pursuing polygonal privacy - about how to fence off a yard with the minimum amount of fence
3. Making winning connections, about the game of Hex, which has very simple rules, but a lot of strategic depth, and which is a favorite on math campuses around the world
4. Jumping champions - about patterns in the prime numbers (that is, integers greater than 1 that are divisible only by themselves and 1)
5. Walking with quadrupeds - about the gaits of different 4 legged animals
6. Tiling space with knots - about some very odd ways of tiling a floor, or a volume of space
7. Forward to the future I
8 Forward to the future 2
9 Forward to the future 3 - which are, as you might guess, the exception to the "stand alone" rule I mentioned above, and are about the possibility of time travel.
10. Cone with a twist - about sphericons, which are unusually shaped objects with some odd properties
11. What shape is a teardrop? - The classic shape of a teardrop, used in countless illustrations, is wrong.
12. The interrogator's fallacy - about some problems in probability (see below)
13 Cows in the maze - about mazes that are actually puzzles in logic
14. Knight's tours on rectangle - the title says it
15. Cat's cradle calculus challenge - about the mathematics of the string game cat's cradle
16. Glass Klein bottles - Klein bottles are sort of 3 dimensional version of Mobius strips
17. Cementing relationships - about some connections between math and art
18. Knotting ventured, knotting gained - about knot theory with real string
19. Most-perfect magic squares - about some special magic squares (see below)
20.. It can't be done! - about mathematical impossibility (see below)
21. Dances with dodecahedra - about some connections between mathematics and dance
Each chapter ends with a summary of reader feedback, websites for more information, and articles and books for further reading
Some of my favorite parts of Cows in the Maze
First, I'll note that these are entirely personal. These are chapters that interested me most - the others aren't of lower quality - indeed, the quality is uniformly high.
In chapter 1, Ian Stewart covers some curious aspects of dice. Here's one that I liked. The notion of transitivity is basic in math. The relation "is greater than" is transitive on the real numbers. That is, if A is greater than B and B is greater than C, then A is greater than C. If we take ordinary six-sided dice and paint different numbers on different faces, we can sum the totals of the faces, and those totals will be transitive. But if we were to roll dice, and bet on which die will have a bigger number, that is NOT transitive. That is, it's possible for die B to be better than A, and C better than B, but A better than C. How is this possible? That would be telling. But it's true, and it's very counter-intuitive.
In Chapter 12 Ian Stewart discusses probability. I am a statistician by trade, so this is near and dear to me. Two of the conundrums he covers are these:
A. The Smiths have two children. One of them is a girl. What is the probability that the other is a girl? Again, telling you the answer would be telling. But it is NOT 50% (even if we ignore the fact that boys and girls are not exactly equally likely to be born, and ignore issues such as identical twins).
B. It's clear that some confessions to crimes may be false - for all sorts of reasons. But might it be that a person who confesses to a crime is LESS likely to be guilty than one who does not? Here, I won't spoil it if I tell you that the answer is "yes".
In Chapter 19 Ian Stewart covers some very special magic squares. A magic square, as you probably know, is a square array of numbers where each row, column and diagonal adds up to the same number. But here are magic squares with a great many more qualities. He also shows how some very big numbers get into math. For instance, if the square is 12x12, then there are about 2.22953*10^10 different "most perfect magic squares". The exact number is known, but, of course, no one will ever list them all.
In Chapter 20 of Ian Stewart discusses the notion of mathematical impossibility, and how it is possible for mathematicians to know that something doesn't exist.
If the above review piques your interest at all, you will probably really enjoy this book. For people who like math, this is a treasure trove. It could also be very useful for math teachers, who want to spice up their lessons with some really interesting ideas.
About the author: Ian Stewart may be best known to the general public as the current contributor of the Mathematical Explorations column in Scientific American. But he started writing books about popular math long before he joined Scientific American, and he also writes novels (three are listed in the frontispiece) and books about Discworld (another three). This is his 29th book. He's also a research mathematician and a professor of mathematics at Warwick University.
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