An Approach to Problem Solving

Told through fables

I have observed that when faced with a problem or task, determined people tend to begin attempting to solve it immediately. They propose hypotheses, argue their points furiously and get to work. This immediacy often times leads to revisiting old ideas and dead ends. The purpose of this post is to provide a few simple steps to take before attempting to solve a problem, each conveyed through a short story.

One day, a rich man lost his precious pocket watch. The watch had been passed down in his family for generations, and he was desperate to find it. He told his servants he would give the first to find it a gold coin. The servants raced to find the watch. They searched for hours, but were unable to find the watch anywhere in room. Just when the man thought all hope was lost, one of the servants who had not even attempted to find the watch approached him. He claimed he would find the watch if everyone else stopped looking. The man agreed, and stopped the search. The servant closed his eyes, paused for a few seconds, and reached into the man’s coat pocket and pulled out the watch and a gold coin. Impressed and , the man asked the servant how he had known the watch was in the coat when he himself had forgotten. The servant said that once everyone else was quiet, he was able to hear the ticking of the second hand and could therefore easily find the watch.

Before taking any action, ensure that you have collected and understood all relevant information. A few seconds of careful thought can save hours of undirected searching.

2. Once a mercenary was traveling through the woods on the way to a battle. In the hollow of an ancient tree he saw a bow. Try as he might, the mercenary was unable to dislodge the bow. A wise man approaches him from behind the tree and says “this all infallible weapon was created to protect the weak. Only he who swears oath to defend weak will be able to procure the bow.” Unable to resist the allure of an all-powerful weapon, the mercenary swears the oath and the bow falls from the tree into his hands. When the mercenary arrives at the battle, the bloodshed has nearly come to an end. In favor of peace and fearful of the bow’s power, the wise man attempts to convince the mercenary to not participate, but the mercenary is determined to fight. The wise man then tells the mercenary “With your weapon you are invincible, therefore, whichever side you join will be the more powerful one. By the oath you swore, you cannot take the bow into battle.”

Always rigorously define parameters before undertaking any task. A statement in which the terms are not absolutely clear is meaningless and clouds understanding.

3. The city of Königsberg is famous for its seven bridges which connect two islands to mainland:

Bridges of Königsberg

An open question was whether one could find a path through the city which crossed each bridge once and only once. This seemed to be a very hard problem, since one could try many paths through the city. The mathematician Leonhard Euler developed an elegant solution to prove that no such path exists, by representing the each landform as a node, and each bridge as an edge between the nodes:

Bridges of Königsberg, graph representation

By using this model, in mathematics called a graph, Euler was able to solve the seemingly complicated problem of finding paths through the city by solving a much simpler problem: under what conditions is there a path through a graph which traverses each edge exactly once? He proved the condition is that there must be either zero or two nodes for which the number of edges is an odd number. Since this particular graph has three nodes which have three edges (all nodes representing the non-center island), no such path exists.

When a problem seems complicated, use abstraction to build a model which represents only the relevant components of the problem. The model will allow one to solve an equivalent, simpler problem.