Common Types of Problem Solving

Here's a puzzle for you. Look at the matchsticks in the picture below. They've been arranged to form an equation in Roman numerals: Vl = ll. But obviously six doesn't equal two. Your job is to move one of the matchsticks into a new position to create a correct equation.

VI = II

How would you start trying to solve this puzzle? Here are some of the ways people go about solving problems.

TRIAL AND ERROR: Before Thomas Edison found that a carbonized thread would work as a filament in his light bulb, he tested hundreds of materials. Even animals use trial and error methods randomly, but skilled problem solvers use this technique in a methodical fashion. They start by ignoring possibilities that obviously won't work. Edison, for example, didn't waste time trying to use cooked spaghetti as a filament. As problem solvers eliminate each of the more likely possibilities, they are left with fewer and fewer places to look for the correct answer. Mechanics call this process "troubleshooting." That's a shorthand way of saying, "I've got a mental list of all the reasons why this machine usually breaks down. Now I'm going to check out each of them until I find out what went wrong this time."

You can try trial and error when you tackle the matchstick puzzle. That means moving each matchstick to every possible position and evaluating the results. If you have enough patience this type of brute force approach will eventually yield a solution—but only if you're alert to unexpected possibilities.

INDUCTIVE AND DEDUCTIVE REASONING: To reason means to reach a conclusion by systematically collecting data and thinking through a problem or situation. Inductive reasoning starts with specific cases and then looks for ways to discover a principle or make a generalization. Suppose, for example, that every time you see a horror movie you have trouble sleeping for the next few nights. Since you normally sleep like a log, you can reason inductively and make the generalization that watching horror movies interferes with your sleep. Similarly, a researcher might use inductive reasoning while testing the effects of overcrowding on rats. If the rats respond to overcrowding by developing signs of neurotic behavior, such as fighting and not caring for their young, the researcher can induce a principle: Overcrowding causes disturbed behavior in rats.

Deductive reasoning, by contrast, uses general principles to provide a prediction or insight into specific cases. You might deduce, for example, that going to see a comedy will not interfere with your sleep because you've gone to funny movies many times before and never had any difficulty sleeping afterwards. In the same way, if a scientist has taught numerous pigeons perform a certain trick, she can deduce that a new pigeon will learn to do that trick too if given the same training.

Will either type of reasoning work with the matchstick puzzle? You can deduce that obvious solution won't work, or it wouldn't be much of a puzzle. For that reason, you can stop trying to make the VI into II/ 1. Remember, you're only allowed to move one matchstick.

INSIGHT: Have you ever experienced an "Aha" reaction? Perhaps it happened the last time you were stumped by a problem. You tried one approach and then another, but nothing budged. Then, suddenly, a great idea popped into your head. You said "Aha!"—and happily put your solution to work.

That's insight, the sudden perception of key relationships that leads directly to a solution. Psychologist Wolfgang Kohler demonstrated that animals are capable of using insight to solve problems in an experiment with a chimpanzee named Sultan. Kohler gave Sultan a stick, then dropped a banana outside the cage. Sultan soon learned to use the stick to rake in the banana. But then Kohler changed the problem by giving the chimpanzee two shorter bamboo sticks. Only if Sultan fitted the sticks together would he be able to reach the banana this time. Sultan tried one stick and then the other. No luck. Frustrated, he took the sticks and sat in a corner. After a while Sultan seemed to have had an idea. He slipped the smaller stick into the hollow end. Then, holding the longer tool he had made, Sultan raced to the side of the cage and raked in his prize.

As Sultan discovered, insight usually comes only after some hard thinking. However, insights often come when you're relaxed, when you have "turned off" your directed thinking for a while. One good rule when searching for insight on a problem is to "sleep on it." Review every aspect of the problem just before you drift off, thus guiding your unconscious to work while you sleep. In many cases, the solution will appear in a dream or will be waiting in your conscious mind when you wake up.

Now have you solved that matchstick puzzle yet? The people who do solve it report that the solution usually comes through insight. Here's a hint: what if the answer required the use of mathematical signs as numbers?