To Guess Reasonably

In the last column, I gave an example of a puzzle that could be solved easier if one assumes that it does have a simple solution than if one works it through from scratch. But the world is full of counter-examples where one goes terribly wrong by assuming the puzzle is well-posed and thus has a simple solution. Sometimes these are obvious begging of the question such as the classic “Have you stopped beating your wife?” This is obviously not a well-posed problem since it presumes a situation which might not be true. Therefore, even attempting to answer it is a mistake. Of course, if you are asked this question in a courtroom and refuse to answer, you are just as bad off as if you did. That’s when you want a good attorney to holler “Objection!”

Other typical, not-well-posed questions that we encounter in everyday life are not always so obvious. Such as, “What are you doing to save your soul?” The issue here is at least threefold: what is a soul, what does it need to be saved from, and can you voluntarily do anything to help save it? I use this only as an example of serious issues that people can fight over without thinking them through. Do not assume I’m taking a position on any of those three positions. However, some respected people have certainly taken opposite sides on almost any combination of answers you can come up with. If everyone would stop and look at the question to see if it is valid before trying to find an answer, then we would likely have less strife.

Many pseudo-paradoxes can be generated by making this type of assumption. For some reason, several of them are associated with simple geometry. For instance, there is the hoary old proof that all triangles are isosceles. This is available at many sites on the Internet, but requires drawings, so I won’t reproduce it here.

When to make assumptions and when not? Several years ago I found two slims books at a garage sale. They are a two volume set entitled Mathematics and Plausible Reasoning by Polya. The title was enough to hook me, but the subtitle of the second volume did me in: Patterns of Plausible Inference.

Isn’t that what we are all looking for? Patterns of plausible inference are what we build our lives out of. With such a great title, I feared the text would not live up to the challenge, but it did, and that slim two-volume set is still one of my favorites. In essence, Polya tries to teach us to guess reasonably. If you can guess and answer and then go back and prove your guess is correct, you can save a lot of effort. The alternative is to attack directly, pushing forward bit by bit.

Of course, you can guess the answer to a problem and rigorously prove it to be correct, but that does not ensure that you have found all the possible answers. You only found one of them. Still, even if your problem has an infinite number of answers, it sometimes helps to know at least one example of a satisfactory answer before setting off to find the rest.

How many universities offer classes in creative guessing? Back in the dark days when slide rules were the only calculating tools commonly available, practitioners became proficient in estimating the result of computations so that they could place the decimal point in the right place. This was a good skill to have, and is still useful to check if Excel has been set up correctly.

BTW, for those of you who do not know what a slide rule is or why the answers it gives have no decimal point, and for those of you who do know, but want a hoot; here is a site where you can download a slide rule to use on your computer http://www.versiontracker.com/dyn/morein… . It even allows you to select different historical models. You use the mouse to slide it. If you are old enough, just the existence of such a thing can make you laugh.

Maybe next time we can consider the book Innumeracy.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.