WHEN MATHEMATICS GETS FUNNY

Two mathematicians were in a restaurant. One of them was a hard-core misogynist and claimed that women were never any good at maths, especially the blonde ones. His friend claimed that there was no difference and that women were just as capable as men. When the misogynist went for a cigarette, the other guy called over the blonde waitress.“My friend and I are having an argument. When he gets back I’ll call you over and ask you a question. The answer you need to give is ‘a third X cubed’. Can you do that?”“Thurdeks coobed?”“a third X cubed.”“a Third Ekscubed. Sure I can do that.”The other chap comes back to the table and his friend says.“I’ll prove to you that women are as good as men at maths. See the blond waitress; I’ll ask her a question and we’ll see if she knows any maths.”So he calls the waitress over and asks, “What is the integral of X squared?”As quick as a shot, she comes out with “A third X cubed.”The misogynist is stunned. The waitress smiles and walks away.Then she stops and calls back, “plus a constant.”

x

                                            LEARN TO THINK LIKE A COMPUTER SCIENTIST

“Once I learn how to solve a complex problem elegantly, i can apply that algorithm fine, but how does one train themselves to see these problems differently?”

This is actually pretty straightforward. Let’s walk through an example.

You’re given an i×j$i\times j$ grid of dots. Starting from the top-left dot, how many unique paths are there to the rightmost bottom dot, assuming that paths can either go down or to the right?

If I give you the answer, I have no doubt you would understand it. But you’re asking about how to get to the answer in the first place.

First, look for a simple & stupid solution. Then begin repeatedly iterating over questions like these:

1. What this this problem like? (Is it a graph theory problem? A linear programming problem?)
2. What other problems have I run across that are like this?
3. What is a simpler version of this problem? (What if j=1$j=1$? What if i==j$i==j$?)
4. Can I get any insight from making the problem more complicated? (What if this was an n$n$-dimensional grid?)
5. What’s for lunch? (Take a break and do something else for a while. Your subconscious has plenty to work with at this point.)

Iteration is the key. Unless you’re lucky (or good), the first several approaches aren’t going to work.

Part of the skill you’re looking for is just learning more problems and more techniques. If you’ve been exposed to combinatorics, for example, you’re going to have a much more diverse set of metaphors to throw at this problem.

There’s a psychological aspect as well that’s harder to teach. Lots (many, many lots) of people will try a couple of approaches and give up. Maybe they’re bored, maybe they’re embarrassed that there initial guesses weren’t very fruitful, but for whatever reason, they’re stuck.

What success I’ve had as a researcher is mostly due to being able to generate one bad approach after another without getting (too) frustrated. If you can come up with a dozen somewhat plausible ways of attacking a problem, one of them is probably going to give you enough insight to be able to move forward. But to do that, you have to be comfortable with the fact that at least a half-dozen of those ideas are going to be deeply stupid.

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 With the exception of the Poetics tutorial I took as an undergrad, it was the most life-changing class I’ve taken. He has set up a separate website (Scientific Problem Solving) for people who are not affiliated with the university. It might or might not be useful. I got a whole lot from the class interaction and being able to break down, step by step, how I was thinking about problems. You might want to check out the syllabus for additional reading material.