Simple, Natural, or Elegant Often Means Progress

Wishful Thinking

One seemingly trivial yet useful strategy for doing math or problem solving in general is wishful thinking.

Wishful thinking is when you simplify/change the problem to another problem that you wish it looked like so you could solve it easily. After getting your "wish", you try to build on your evolved form to answer the original problem.

The MAA describes wishful thinking by this question in this article: "If you could change the question, what change would make it ridiculously easy?"

Example of Wishful Thinking

A simple example in which you could use wishful thinking would be to factor $x^4+x^2+1$. It's a fourth degree polynomial, which is hard to solve, so you wished it was a quadratic because quadratics are easier to solve. But wait! It's an even polynomial, so the polynomial could be rewritten as a quadratic: $(x^2)^2+x^2+1$.

If you define $y = x^2$, you can rewrite the polynomial as $y^2+y+1$. Guess factoring leads nowhere, and you could only wish that it was a perfect square. Aha! We could force the polynomial to contain a perfect square by rewriting the expression as $(y^2+2y+1)-y = (y+1)^2 - y$. Since $y=x^2$, you can use difference of squares to factor it.

Here is the complete solution:

Let $y = x^2$.

$\begin{array}{lr} \:\:\:\:\:\: x^4+x^2+1 \\ = (x^2)^2+x^2+1 \\ = y^2+y+1 \\ = (y^2+2y+1)-y \\ = (y+1)^2 - y \\ = (y+1)^2 - x^2 \\ = (y+1+x)(y+1-x) \\ = (x^2+x+1)(x^2-x+1) \end{array}$

More Resources

For more resources on wishful thinking, you can watch this video by the Mathematical Association of America here, or read The Art and Craft of Problem Solving (Chapter 2) by Paul Zeitz.

Imagine you are working against the clock at a math competition, trying to assemble the ideas you need to tackle a hard problem. Possibly, you will imagine the sadistic proposer who dreamt up the hardest problem ever. However, it is more likely that even this problem is quite approachable, and that some contestants will manage to solve it.

With a contest paper, one can look into the head of the committee who compiled it. They will have probably selected problems with fairly simple, clean solutions, because otherwise, nobody could solve them. To find this simple solution, you should consider simple ideas.

You will also find that it is much easier to work with more natural structures, as opposed to convoluted objects, or complex casework. It makes it easier for you to grasp the logic behind it and to come up with new results. Moreover, for this reason, the object of mathematical pursuits is usually quite elegant and comprehensible.

For all these reasons, in your blind gallops across mathematical prairies, natural observations are a good reassurance that you are on the right track.