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# Problem

A **problem** is a situation preventing something from being achieved. The word comes from a Greek word meaning an "obstacle" (something that is in your way). Someone who has a problem must find a way of solving it. The means of solving a problem is called a "solution".

## Contents

## Examples

"John has locked his car keys inside his car so that he cannot get at them. John has a problem".

### Entertainment examples

Some problems are made up for fun. These are like puzzles. Some of them can be solved with logic, others can be solved by trial and error or by a *heuristic*.

### Mathematical examples

Here is an example of a mathematical problem: "John is three times as old as Mary. In three years time he will be twice as old as Mary. How old are John and Mary?" The answer to this problem is that John is 9 and Mary is 3.

Children often like to give one another problems that can be solved by "lateral thinking". This means using the imagination rather than strict logic. Here is an example:

"Peter, Ruth, Samuel and Jessica live in the same house. Peter and Ruth went out. When they returned they found Jessica lying dead, surrounded by glass. They were sure Samuel had done it. Why did they not call the police?" The answer is: Samuel was the cat and Jessica was a goldfish.

A well-known example of a problem that cannot be solved is that of squaring the circle – how to get a square that has the same area as a given circle, using geometry.

## Sources on problem solving

These are classic works on problem-solving:

- Duncker, Karl 1945.
*On problem solving*. APA Psychological Monographs**58**. - Gordon, William J.J. 1961.
*Synectics: the development of creative capacity*. New York: Harper. - Polya, George 1945.
*How to solve it: a new aspect of mathematical method*. Princeton University Press. - Wertheimer, Max 1959.
*Productive thinking*. New York: Harper. New and expanded edition was 1945. - Simon, Herbert A. & Newell, Alan 1972.
*Human problem solving*. Englewood Cliffs, NJ: Prentice Hall.

Most of these deal with well-formulated problems. There are also a number of books on the psychology of thinking, which is obviously a large part of problem-solving:

- Bolton, Neil 1972.
*The psychology of thinking*. London: Methuen.ISBN 0-416-70450-6 - Wason P.C. & Johnson-Laird P.N. 1972.
*Psychology of reasoning: structure & content*. London: Batsford.ISBN 0-7134-0347-0 - Novick L.R. & Bassok M. 2005. Problem solving. In K.J. Holyoak & R.G. Morrison (eds)
*Cambridge handbook of thinking and reasoning*. Chapter 14, pp. 321–349. Cambridge University Press.

## Related pages

- Monty Hall problem – a problem about goats and a car in a game show, using probability.