A couple of months ago, my stepson told me that he was reading a math book. The book had been recommended by The Economist magazine. He told me he would pass it along to me when he was finished, since he thought I would enjoy it. He was partially right.
When he finished reading the book, he explained that he could not grasp all the math in the book but thought it was a decent read anyway. My stepson, I should tell you, is a very smart guy, an anesthesiologist at one of the hospitals out here on Long Island. So, his disclaimer put me on high alert.
The problem I have with math books is this: If I don’t quite understand what the author has presented, I keep re-reading and working on problems until I fully understand what the author is saying. This takes time, and I don’t have that much time to spare. I’m sure many of you approach your technical reading in the same way and often wind up spending way too much time figuring out how a circuit is working or what an equation really means.
In any case, I started reading this book, The Number Mysteries by Marcus du Sautoy. The sub-title is: A Mathematical Odyssey through Everyday Life. This is one of those math books that should truly be an easy read.
I was doing okay in the first chapter: The Curious Incident of the Never-Ending Primes. The chapter refreshed my memory and told me about the latest research in prime numbers and other things. I noted how the author tried to push readers to his own and other web sites for more information and even made it easy to do by sprinkling QR codes throughout the chapter.
But I started to feel agitated while reading the second chapter called: The Story of the Elusive Shape. The thing that bothered me most was that the author talked about complex shapes but the visuals were simply gray scale line drawings on the page. One of his web site recommendations was the Wolfram MathWorld web site, which has all the Archimedean solids. But this wasn’t much better, just color versions of what was in the book plus a bunch more drawings. What was I looking for? I wanted to see a 3-D representation of these and other shapes in the book, something I could spin around with a move of the mouse. I didn’t go looking for these myself, I just continued reading the book.
Chapter 3 was when I stopped reading. The chapter is called: The Secret of the Winning Streak. I was a few pages into the chapter when I came to a sub-head called: How Good Are You at Randomness? Essentially, the author posed the following question: What is the probability of getting a run of three heads or three tails if you toss a coin ten times? He gave the answer as 846/1024 or about 82.6%. The trouble started for me when he explained how he got this number. Not only did I not like his explanation, but I also thought it would not be extendable. I started to wonder how to find the probability of a run of three heads only (or three tails only). I also wondered what the probably would be if you thought of the coin flips in terms of triplets, meaning that if you tossed the coin ten times and got all heads, you would have a run of three heads not once, but eight times, i.e. flips 1-2-3, 2-3-4, 3-4-5, 4-5-6, 5-6-7, 6-7-8, 7-8-9, and 8-9-10.
Here’s how the author explained how he got his 82.6% number. He starts off by saying, “Strangely enough, the Fibonacci numbers we met in Chapter 1 are the key to working out the chances…”
As you probably realize by now “strangely enough” did not sit well with me. He goes on to explain that if you toss a coin N time, there are 2N ways the coin could land—very logical, I agree with this. Then he gets funky and says that if gN is the number of combinations with no runs of three heads or tails, you can calculate gN by using the rule for the Fibonacci numbers: gN=gN-1 + gN-2. This did not make any sense to me. The author then goes on for another couple of paragraphs to explain his reasoning, but it still didn’t make sense to me.
My question to you is this. Knowing this much about the problem and how to solve it, can you get 82.6% as an answer? If you can, then what is the answer to my question about “three heads only” rather than “three heads or three tails” and how does the original solution suggested by the author apply to this solution. I spent a considerable amount of time coming up with solutions to the “three heads only” problem as well as the triplets problem. Of course, I didn’t want just the solution, but the formula too and a sensible way of explaining it to my stepson.
If someone out there tells me that there are simple solutions to these problems and agrees that the author’s explanation was right on the mark, I’ll be embarrassed for sure and probably stop reading math books altogether, since they just take up too much of my time. If not, I’ll continue to plow through this one, looking for more incongruities.