Cure Your Problem Solving Illness: Learn Simple Solutions

[ShayanJ]

I don't know the name of this but it seems that I have some kind of a problem solving illness. It has happened many times in my life that my close friends make fun of it from time to time. The problem is, sometimes when I think about something, its not the simplest solution that comes to my mind and its the more complicated solutions that comes to my mind first!
For example today someone asked me how to prove that the degeneracy of hydrogen atom energy levels(not considering fine structure) is ##n^2##. The solution is to compute the sum ## \displaystyle \sum_{l=0}^{n-1} (2l+1) ##. And this is what I sent him:
## \displaystyle \sum_{l=0}^{n-1} (2l+1)=\sum_{l=0}^{2n-1} l-\sum_{l=0}^{n-1} 2l=n(2n-1)-n(n-1)=2n^2-n-n^2+n=n^2 ##
But, there is an easier way to do the sum:
## \displaystyle \sum_{l=0}^{n-1} (2l+1)=2\sum_{l=0}^{n-1} l+\sum_{l=0}^{n-1} 1=n(n-1)+n=n^2##.

What is this illness? How can I cure it?

 

[Borek]

Don't worry, you are not the first one to have this problem.

The following problem can be solved either the easy way or the hard way.

Two trains 200 miles apart are moving toward each other; each one is going
at a speed of 50 miles per hour. A fly starting on the front of one of
them flies back and forth between them at a rate of 75 miles per hour. It
does this until the trains collide and crush the fly to death. What is the
total distance the fly has flown?

The fly actually hits each train an infinite number of times before it gets
crushed, and one could solve the problem the hard way with pencil and paper
by summing an infinite series of distances. The easy way is as follows:
Since the trains are 200 miles apart and each train is going 50 miles an
hour, it takes 2 hours for the trains to collide. Therefore the fly was
flying for two hours. Since the fly was flying at a rate of 75 miles per
hour, the fly must have flown 150 miles. That's all there is to it.

When this problem was posed to John von Neumann, he immediately replied,
"150 miles."

"It is very strange," said the poser, "but nearly everyone tries to sum the
infinite series."

"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

 

[micromass]

I don't see the problem. A solution is a solution.

 

[ShayanJ]

I don't see the problem. A solution is a solution.

Well, yes. But maybe the example in the OP was not a good example for what I mean. Sometimes it becomes conceptual and I think about a problem in a way that makes things unnecessarily complicated and so harder to think about and solve. And then my friends point out that I'm over-complicating things and explain the simpler way to think about it that gives the answer right away. It feels bad!

 

[micromass]

Well, yes. But maybe the example in the OP was not a good example for what I mean. Sometimes it becomes conceptual and I think about a problem in a way that makes things unnecessarily complicated and so harder to think about and solve. And then my friends point out that I'm over-complicating things and explain the simpler way to think about it that gives the answer right away. It feels bad!

You shouldn't feel bad. Finding the most elegant and short solution is very nice, but it's unrealistic. The first person to prove a given theorem in mathematics almost always had a very complicated solution. This solution is then usually simplified over time. Sometimes an argument of 50 pages gets simplified to a proof of half a page. So this kind of thing happens all the time. But the one who found the long and complicated solution gets the credit!

You know, problems that have an elegant and quick solution usually appear only in textbooks. When you come to real research problems, then it is exactly people like you who get to solve the problem quickly. People who see the most elegant way have a harder time because usually there is no elegant way.

But sure, if you're interested in seeing the more elegant and easier way of problems, then you should solve many problems. It's an experience thing mostly.

 

[tionis]

What is this illness? How can I cure it?

You are too brilliant. You have moved beyond solving problems and should instead start creating them.

 

[Borek]

You have moved beyond solving problems and should instead start creating them.

He just did, have you read his first post?

 

[tionis]

He just did, have you read his first post?

Those aren't new problems lol. He's using what other created for him. I'm talking about inventing new maths.

 

[ShayanJ]

Those aren't new problems lol. He's using what other created for him. I'm talking about inventing new maths.

I think he means I just created a problem for myself by assuming that what I described is a "problem solving illness"!
He's right!

 

[Fervent Freyja]

You don't have an illness, you have the ability to focus and an inner need to make sure you understand the problem thoroughly. Why short-cut the most enjoyable part of solving? You are more than fine by my book.

 

[Pepper Mint]

You are just being a little *unusual* in solving problems, and I think it is your hyper intuition. You know the means to get you to your final goal and your chosen path is far different from what other ordinary people would probably follow.

 

[tionis]

I think he means I just created a problem for myself by assuming that what I described is a "problem solving illness"!
He's right!

Yes. An illness worth having but sadly not contagious