First things first

Once upon a time I was summoned for an interview with GCHQ about a summer internship. Given that I have never done a summer internship with GCHQ, you can already tell that this interview did not go well. However, I learned something useful from this interview about problem-solving which I would like to share with you.

We find ourselves in the interview room, midway through the interview. It has already gone badly by this point, when the interview asks: "Suppose I have a stick which is one metre long, and there is some number of ants on the stick. Each ant moves with a speed of one metre per second and, when two ants collide, they bounce off one another and move off in opposite directions with speed unchanged. How long before they all fall off?"

He is testing my mathematical skills, I thought. There will be some fancy combinatorial formula for this. Suppose there are n ants. Then...ummm...they will be walking along...and ummm...some will fall off...and there should be a formula for it...maybe we should say there are k ants...wow these ants are fast...there should be a formula...

After a little while, he kindly put me out of my misery. "What happens if there is just one ant?" Ah well, he would fall off after one second (at most). "And if there were two?" Well, the longest they could take would be if they set off from opposite ends and then it would take half a second to reach the middle and another half-second to fall off the ends, so one second. Hang on a minute...

So the formula I was groping for was: 1.

The beautiful answer (which I still didn't spot at this point) was that two ants bouncing off one another is indistinguishable from two ants passing through one another, so it doesn't matter how many ants there are, it's always one second.

It didn't help that, when I told the problem to a friend of mine, his immediate answer was "but bouncing off is the same as passing through, so they take one second".

The moral here is: don't always look at the general case first. You might not see the simple answer because of the extra complications clouding the problem. Look at the simplest case first, and gradually add to the complexity, until you spot the pattern.

I still use this insight to help me solve problems to this day.