Does perseveration = linear thinking = difficulty with math?

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I've always found solving math problems really hard because more often than not I'll start thinking something through and get so focused on one train of thought or one specific part of the problem that I may never realize that other solutions are in fact needed. What it means is that something obvious to someone else, like breaking up a sum or redistributing it and then solving other parts of an equation, just never occurs to me.

I've found some of the best people at math are those who can do "numbers gymnastics" or who don't get bogged down in linear perseveration like I do.

Is this something that anyone here can relate to?

In general, once you get into really advanced math, numbers and calculations cease to have much meaning except as examples in very limited cases. Everything else is so abstract that the numbers and calculations are left far behind.

There are a couple of stories about a famous mathematician named John von Neumann. (Note that there are variations about the stories but the stories are substantially the same. Also, these are from recollection from more then thirty years ago.)

One story involved a puzzle about a fly and two bicycles. Two bicycles (with bicyclists on board) start out 20 miles apart, and each travels toward the other at 10 mph. At the instance they start, a fly takes off from the front of one bicycle and flies toward the other at 15 miles per hour. When it touches the second bicycle, it immediately turns and flies back toward the first at 15 miles per hour. It repeats this over and over again until the two bicycles meet.

Someone gave this problem to von Neumann and he replied immediately with the correct answer. The questioner said something like, "Oh, you must have already heard this and know the trick." von Neumann answered, "What trick? I summed the infinite series."

The usual initial approach and what von Neumann had done was to determine how far the fly travels on each leg of the trip and then sum up the individual trips to get the total of 15 miles. It's not a terribly hard problem but for most people who are well versed at doing such problems, it would take at least a couple of minutes to do.

The trick is to note that it takes the two bicycles one hour to meet and thus a fly traveling at 15 mph for the hour will cover 15 miles. Thus, you don't need to sum an infinite series to figure out the answer.

Another story occurred during World War II. On one of his trips to the University of Chicago from the Manhattan Project, von Neumann was introduced to a graduate student and told of a problem the student was working on.

The graduate student had just completed figuring the solutions for the simplest three cases of the problem. For the simplest case, n=1, it only took the graduate student a couple of minutes. For the next case, n=2, it took several hours. And for the third case, n=3, it took much or all of the night.

So von Neumann thought a few seconds and gave the solution for n=1. Another minute or so and he gave the solution for n=2. For the third case which was really quite difficult, after three or four minutes of thought, the graduate student blurted out the answer because he didn't think von Neumann could solve it in his head. von Neumann just gave him a dirty look and went back to figuring out the solution. After another two or three minutes, he looked back at the graduate student and said that the graduate student was right. von Neumann was amazed that the graduate student was able to figure it out so fast.

There is at least one other variation of this story and I don't know which, if either, is correct.

At first I thought the answer was 2 trips because I didn't read the question correctly. Then I thought it was the bicycles never met because I thought they were going the same way. Only when I read the answer I realized they were going towards each other but it was too late to work the answer out because I already knew it.

As for solving Maths questions in general I need to write them down along with all my working out so I don't lose track of where I am and what my final goal is.

If the fly never stops when it turns around and is flying a steady 15 mph find out how long it takes the bicycles to meet up and multiply that time by 15 mph.

ruveyn