## Wednesday, February 8, 2012

### How the Other Half Thinks: Adventures in Mathematical Reasoning How the Other Half Thinks: Adventures in Mathematical Reasoning
by Sherman Stein
Published in 2001
ISBN: 0-07-140798-7

Last year I read and reviewed John D. Barrow's excellent 100 Essential Things You Didn't Know You Didn't Know. Barrow's book introduced the general public to 100 math problems and general applications which he related in an engaging and interesting manner.

Sherman Stein's book belongs to the same general genre. Writing for a layman with no specialized training, Stein introduces a handful of problems that mathematicians have tackled through the years, and methodically and step by step he takes the reader through the logic inherent in solving the problem, or proving the proposition.

Barrow's book was divided into 100 sections, each looking at a different aspect of mathematics. In contrast, Stein deals with just eight. In each section, he poses a question, then guides the reader through the logic used by mathematicians to explore it. I deliberately use the word 'explore' rather than 'solve'. Correct answers are almost an afterthought -- the process is the thing.

This book is about logical thinking more than math -- not to imply there's a line where math ends and logic begins! The reader isn't assumed to have any mathematical knowledge beyond how an equation works, how an average is calculated, or what pi is.

In the very first chapter, Stein gives us what appears to be a difficult problem. Imagine you drop a needle onto a floor, and the floor has a pattern of parallel lines which are evenly spaced apart. What is the probability the needle will cross one of the parallel lines? As you can probably guess, that depends on the relationship between the length of the needle and the distance between the lines. (This is known among mathematicians as the Buffon's Needle problem.)

Next, he makes it really interesting.

What happens if we keep the length of the needle and the distance between the lines constant, but we bend the needle? Bend it into a right angle, bend it into a W, bend it into a rectangle, it doesn't matter. How much less likely is it that the needle crosses a line?

The answer -- it doesn't affect the probability at all! -- is counter-intuitive (at least, it was for me), but it is impeccably proven, and proven in a way that doesn't even use what we would consider math.

The remainder of the book covers topics such as infinity (containing the famous old question of whether one infinite set can be larger than another), streaks (concerning the tendency of random processes such as a coin flip to yield surprisingly long streaks if given enough time), and letter strings (in which the strings are ingeniously visualized in two dimensions as road maps).