Reply #16: Here's the answer from the Paul Zeitz DVD

[pokerfan]

ETA: fixed some typos.

I made some screen grabs for illustrative purposes. It's from this video series by Paul Zeitz but was originally posed years ago by Martin Gardner, Scientific American.

http://www.amazon.com/Art-Craft-Mathematical-Problem-So...

Well worth it. Anyway, while I couldn't find a clip to share (except on bit torrent) I managed to find a transcript. The names were changed to foil the googlers...

Now, the question is: Is there even enough data? We don't know the speed that Ernie is going. We don't know the speed that Burt walks. Well, it turns out it doesn't matter, and one way to organize the little data that we have is by drawing a picture. Let's try adding a picture to this problem, and the natural thing to do is a distance-time graph. So here's a graph where the horizontal axis is time; I'm labeling it 4:00 and 5:00. The vertical axis is position; H stands for home, and S stands for the station. Here's a typical day. This is what Ernie does. Ernie leaves some time before 5:00. We don't know when, so I start at an arbitrary starting time. Then Ernie travels in a straight line because it's constant speed from H to S, arriving at S, the station, at exactly 5:00 and then reverses the path. So what you get is something geometrically nice, an isosceles triangle. We don't know the slope of the triangle. If Ernie went fast, it would be very steep. We just don't know that. Now, on the day of the power failure, Burt has arrived at the station at 4:00, and Burt begins walking toward home at a much lower speed than a car, so it's some kind of a shallower sloped line. The green line is Burt's progress, and those red lines are Ernie's. Burt will walk until intercepting with Ernie. Once that occurs, then they head back, going at the car speed. That blue line is the new path, which is parallel to the original red path. That's the picture. So pictorially that's what's happened, and what we know is when that blue line finally hits the bottom of the graph, the time axis, that would mean that Ernie and Burt have gotten home. What we know for sure, is that that is going to be 10 minutes earlier than the normal arrival time. What we need to find out is how much time Burt has walked, and so what we need to know is how much before 5:00 it was when the interception occurred



So let's blow up the picture and look at the top of that triangle. If you look at it, that thick, black line stands for the 10 minutes before the normal time of getting home. We drop a perpendicular down, and it bisects that because it's an isosceles triangle. It bisects that thick, black line, and so we know that's going to be 5 minutes.



Yet if we drop that down, we see that the interception time had to have been 5 minutes before 5:00. So what we deduce is that Burt walked for 55 minutes. This was done without algebra, just done with pictures.



Now, the pictures are a little complicated, but they're fun. The key thing is that we've been able to convert a word problem into the geometry of parallel and perpendicular lines. Now, is it always worth it to ask the question: Can we draw a picture and if so, how? Maybe it's not always worth it, but in general, it's a good idea to look at a problem and think is there some way we can force this into a visual mode because at least it will loosen you up.