^{The mutilated chessboard} |
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## Representation and restructuring

The picture to the right shows a chessboard with two corners detached. Suppose that a domino will cover exactly two squares on the board. Is it possible to take a set of dominos and place them so that the 32 squares on the board are exactly covered in dominos (that is, the dominos mustn't go off the edge of the board or overlap each other)? Explain your reasoning.

This is known as the mutilated chessboard problem (or mutilated checkerboard problem)^{1}. Kaplan and Simon (1990)^{2} investigated performance on four versions of this problem:

- A board filled with blank squares.
- A board with pink and black squares (as shown in the diagram here).
- A board in which the squares were labelled with the words "pink" and "black", instead of the colours themselves.
- A board in which the squares were labelled with the words "bread" and "butter".

They found that performance increased from problems 1 - 4. The correct answer is that it is not possible to cover the board in dominos. Consider the board with pink and black squares. On an unmutilated board there are 32 black and 32 pink squares. Because each domino must cover both a black and a pink square it would be possible to cover the board exactly. However, on the unmutilated board there are 32 black squares but only 30 pink squares. Therefore, it is not possible to cover the board exactly.

A person who simply tries to visualize placing dominos on the board is likely to run into memory difficulties. A better way to represent the problem is to include the exact numbers of black and pink squares, plus the information that a domino will cover one of each. This representation becomes increasingly likely with versions 1 - 4 above. Shifting from a naive representation of the problem to the more sophisticated one can be considered an example of *restructuring*.

## The nature of insight

The solution to some problems seem to arrive out of nowhere, whereas with other problems the solution seems to be the result of a long thought process. There is a debate as to whether an information-processing approach can account for the solutions to both. If problem solving is a process of search that can call upon existing knowledge in memory, then it might be expected that "feeling-of-knowing" judgments would just as readily be produced for insight problems and noninsight problems. In one study^{3} participants were asked to give "ratings of warmth" (i.e. estimated closeness to solution) every fifteen seconds as they worked on problems. With algebra, logic, and Tower of Hanoi problems, there was a gradual increase in warmth before a solution was reached. However, with insight problems there was barely any increase in warmth until right before a solution was reached.

By contrast, Kaplan and Simon's investigation of the mutilated chessboard problem led them to argue that a straightforward information-processing account could explain the results. As indicated earlier, trying to visualize placing dominos on the board is a challenge for limited capacity working memory. Kaplan and Simon argued that successful solution required constraints to be incorporated into people's problem representations. Indeed, think-aloud protocols produced by participants indicated that all began by trying to imagine placing dominos on the board, but then switched to different representations. They all eventually tried to find a solution based on a parity representation - i.e. a solution based on recognizing that each domino had to cover both a pink and a black square.

## Incubation

When people have worked on a problem without reaching a successful solution, they often put it aside for some time with the intention of returning to it later. Sometimes the solution pops into their head during this period when they were not thinking about the problem, or quickly once they have returned to the problem. However, these *incubation* effects have not been widely investigated.

Question: What explanations can you think of for incubation effects?