Sunday, August 12, 2012

Spider Calculates Best Path

Still stalled on physics, so I was fooling around with a math puzzle today. I don't know where math puzzles come from. There are a lot of math puzzles out there but there are only so many really great math puzzles of this particular type. I'm talking about puzzles which look very difficult but which are easily solved by taking just the right approach. Today's problem is of that type. I don't know who invents these problems, or how old they are. They just are.

In today's problem, there is a room with a spider in it. The spider is in the middle of the end wall, two feet from the ceiling. There is an electrical outlet on the opposite wall, also on the centerline of the wall, but two feet off the ground. Suddenly an exterminator enters the room and starts spraying it with poison. The spider panics and tries to take refuge inside the electrical box, thinking it might be safe in there. This is a picture of the room:

And of course, as we expect of insects in math problems, the spider instantly calculates the shortest possible path along the walls, floor and ceiling, and heads for its destination along the calculated trajectory. What is the trajectory?

At first glance, we think the spider will stick to the centerline of the room, going either straight up or straight down. But a little reflection shows that there are other possible paths, and indeed the obvious path is not in general the shortest.

I'm not posting this problem because I intend to tell you how to solve it. There is a marvellous method for easily calculating the shortest trajectories, and you may or may not already know it. If you don't, it's worth figuring out for yourself. That's not what I'm interested in today.

Maybe you notice that I haven't specified the length of the room. You can put in any length you want and it's a perfectly good problem. What not everybody knows is that depending on the length you choose, the problem can have very different solutions.

Unless I'm mistaken there are four possible solutions. I'm not going to tell you what they are, except that in certain conditions even the obvious solution becomes correct...the case where the spider stays on the centerline. But as you vary the length of the room, the nature of the solution changes in surprising ways. What I've done is put the numbers into a spreadsheet and calculate the transition points. I wanted to verify that each of the four shall we say "topological" cases actually occurs, and in fact they all come into play for the right parameters. In fact I didn't need to vary the width and height, only the length. Here are the transition points:

I'll leave it to you to figure out what the Type I, II, III, and IV solutions look like.