Mechanics of Problem Solving

Over the years, I've been trying to seek a decent illustration of what exactly happened in solving an olympiad-level-math-problem. It's not as simple as arranging the chess pieces on the chessboard correctly. But on the other hand, it's not as convoluted as finding a needle hidden among stacks of hay. Problem solving is not completely random or only based on creativity, but it also consists of some routine. Today I want to present something that hopefully, would be able to provide you sufficient insight.

Ever heard of this game : Unblock Me ? It's a puzzle game of moving tiles horizontally or vertically. The goal is to be able to move the red tile out to the right of the board.

An instance of Unblock Me's puzzles

Not until I played 1000+ of these, I realized this game's relevance to problem solving.

Imagine the red tile as the given of the problems. You want to move towards your goal (usually prove .... or find ....) on the right side. But instead of regarding the tiles as obstructions, think about it as the path towards your goal.

The path can be in the form of theorems or lemmas*. As you can move the tiles up and down, right or left, you can utilize theorems in different ways. And each choices you make could be the striking difference between getting closer to the finish line or straying beyond the original problem. Unfortunately, in some cases, it is not clear to which direction you are actually going.

The good news is the more practice you do, the more able you will be to grasp the depth of the problem. One method which I applied often from the problem solving method is thinking backwards.

The 'backward thinking' process

In this game, from the simplest thing as noticing that the top part (marked by green) only has 2 spaces, while the bottom part (marked by yellow) has 3 spaces. So that means the blue-edged tile, which covers 3 spaces, must be pushed towards the bottom in order for the red tile to pass through. Furthermore, both horizontal tiles under the blue-edged tile must be pushed to the left to give the needed space. And so on, and so on, as clearly explained by the blue arrows.

It's the same as solving olympiad problems. The more you practice, the more intuitive you are to find the blue-edged tile and the faster you will be to trace the blue arrows. You would be able to reason for every steps you make. Well, that has always been my definition for 'completely solving a problem'.

The light is near

And by the way, I warn you : do not apply this directly to your life's problems. As much as I want to simplify my life, this illustration is just meant for these kind of problem solving. In real life, tiles can crack and sometimes you couldn't even see any way out. But don't worry, keep up your hope and do what you can do. Pray for the rest, and eventually you will see the light =)

*What is a lemma ? http://en.wikipedia.org/wiki/Lemma_(mathematics)