August 11, 2015

I have another math brainteaser for this week. It's easier than last week's problem.

You're taking a long drive, and you want your average speed for the trip to be 50 miles per hour (mph). But you run into some traffic, and when you've traveled half the distance to the destination you notice that your average speed so far has only been 25 mph. How fast do you have to drive during the latter half to still meet the initial goal of averaging 50 mph for the whole trip?

a) 0 mph

b) 25 mph

c) 50 mph

d) 75 mph

e) 100 mph

f) infinitely fast

Think about it, and choose your answer before you scroll down. Meanwhile, let me show you this graph to take up some screen space, so you won't see the answer right away:

That is a rough graph representing the probability that you will give the correct answer, as a function of the intelligence, education, and effort you bring to the problem. Note the dip in the middle, where you're LESS likely to get the question right than if you were stupid and ignorant. That dip is what I'm calling the intellect trap.

Now, are you ready for the answer?

The correct choice is "f) infinitely fast". To see why, imagine that the length of the total trip is 50 miles. Your goal is to averaging 50 miles per hour, so you're hoping to arrive at the destination in 1 hour. Then the half-way point would be at 25 miles, and it's here that you've realized that you've only been averaging 25 mph. Meaning, you've traveled 25 miles at 25 mph, using up the entire 1 hour that you had allocated for the whole trip. So you have no more time - zero hours - left to travel the remaining 25 miles, and therefore need to travel at an infinite speed.

I once gave this problem to a group of 9th grade geometry students. They knew what "average speed" meant. They knew that "distance = rate * time". They had done well in algebra. They were bright students. A five-year-old would have chosen the right answer 1 out of 6 times by chance. Yet, out of that class of about 30 students, only one got the right answer, and he had chosen his answer as a joke. Most of the class had mistakenly chosen 75 or 100 mph as the answer, which are the seemingly correct values upon a superficial examination. They had fallen for the intellect trap.

That, I think, is the most salient feature of this problem. There are situations where more intellect, education, or effort actually DECREASE your chances of getting the correct answer. And upon a moment's reflection, you'll see that this phenomenon is actually quite common. Consider one of your personal areas of expertise, a subject you know well enough to have taught or supervised others in. Isn't there something like a list of common beginner mistakes, or a "gotcha" moment in the flow of progress one makes?

Now, we would be fools if what we took away from this was "look at these stupid 9th graders, so dumb that even when they try harder it only makes them more likely get the wrong answer". No: what concerns me is "what intellect traps am I falling into? Which of my ideas are only half-baked without me realizing it? What traps out there are so large, that my whole field or segment of society has fallen into it? And how could we tell when we're in one?"

Note that I'm NOT just saying "be careful what you believe", "think about your opinions", or "do some research to back up your positions". All that's just common sense. I'm concerned here about when that common sense FAILS, when more care, more thought, and more research only leads more to the WRONG conclusion. How can we detect or prevent this?

One thing that comes to mind is to beware the feeling of contempt, which exacerbates the issue. Look at the structure of the multiple choices given in the initial average speed problem. The first three options are obviously, contemptibly wrong. If you're already behind schedule, how could you possibly catch up by doing nothing or slowing down? After dismissing these answers as obviously wrong, it becomes easier to also dismiss the last, correct answer this way when it doesn't conform to one's intuition.

A second precaution is to think past where you think you've reached the right answer. This is what I've tried to demonstrate in my solution to the two envelopes problem (which you may want to read first). First, there is the intuitive understanding that your opponent (the wild statistician) must be wrong, because what he proposes (that you continually switch the envelopes) is clearly nonsense. But beware the feeling of contempt: you cannot simply stop here. You must also propose the right way to think about the problem. "It is better to light a candle than to curse the darkness". Next, you should think further until you identify the exact nature of your opponent's error. And ideally, you should then be able to correct their error and redeem your opponent's thought process, so that it reaches the same conclusion as your initial way of thinking. In short, you'll know you've reached the end of this line when your thoughts are a superset of your opponent's thoughts. In doing so, you may be able to identify the intellect trap as the mistake that your opponent made.

This is, of course, not enough to completely guarantee that you're right. Nothing ever is. However, thinking about it from two different perspectives - such as yours and your opponents - and reaching the same conclusion greatly enhances your chances. Three or more perspectives are better still. I leave it as an exercise for the reader to find multiple other ways to solve the initial average speed problem, which I've only solved in a very limited way above.

I also realize that, looking at this post self-referentially, I've tackled this "intellect trap" problem in an incomplete way. I don't yet have a complete solution to which procedure will let you best detect and escape the trap. I only have several rules of thumb. If anyone has the complete solution (say, a complete Bayesian formulation of the problem and the answer), I'd be happy to hear from you.

But meanwhile, I feel that this is a common problem, and especially pernicious and difficult to detect at the society-wide level. With the benefit of hindsight, we can sometimes see when entire cultures fell victim to it, such as when communism gained favor among the intelligentsia of many countries near the beginning of the 20th century. But what such traps lurk in our world today? I worry about the polarization of politics, or the paucity of Christians in academia. These phenomena seem to be driven in part by this intellect trap. In these issues, I unfortunately see a great deal of contempt, few attempts at an improved solution, little attempt at engagement, and virtually no efforts to redeem the opponent's thoughts.

You may next want to read:

Basic Bayesian reasoning: a better way to think (Part 1)

Science as evidence for Christianity against atheism (introduction)

Another post, from the table of contents

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