### Inspiration, Perspiration and Satisfaction

As Cornell mathematician Steven Strogtaz wrote in his recent (and very much recommended) book *The Joy of x* (p. 94),

Without inspiration, there'd be no proofs - or theorems to prove in the first place.

But inspiration is a capricious lady - and I do not mean that in any deprecating manner, just that sometimes it is absent, and sometimes it turns up insights that do not compare favorably with the alternatives.

A case at hand was a diagram by Hubert Shutrick that intended to illustrate a proof without words for Bottema's theorem. Practically by definition, a proof without words ought to be so obvious as to unambiguously convey exactly same explanation and conclusion to various observers. Hubert's diagram

inspired me with a vision of Bottema's theorem and its proof which was not only somewhat different from, but, as it became clear on subsequent communication, inferior to, what Hubert intended.

A second incident happened a few days ago. In an old problem book I was browsing one geometric problem caught my attention.

With eyes closed, I visualized the picture which looked quite suggestive. The problem appeared rather simple and the solution sprung in my mind almost immediately. What a surprise it was to find out that the solution at the back of the book was much simpler, shorter, and, in a sense, more natural. How could I have missed it?

Still, there is great satisfaction in having solved a problem - even a simple one, and extra satisfaction in being able to appreciate an elegant proof; this kind of satisfaction is multiplied manifold after you devised a solution on your own. Yes, it all may start with inspiration, but to keep the flame burning involves hard work. The upside is that eventually it all comes together when, with evolving habit of solving problems, one begins to realize how right is the paraphrase of the well known statement by Thomas A. Edison:

Solving problems is one percent inspiration and ninety-nine percent perspiration.

The more you sweat the greater is the satisfaction.

[...] was inspired to write this because of this post by Alexander [...]

January 29th, 2013 at 1:05 pm