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122 Creativity

Figure 4.7. The Nine-Dot Problem and its solution.

problem allowed us to predict with precision the rank order of the difficulty of
such problems, the effects of repeated exposures and even the distribution of
visual attention over different components of the problem, as well as changes in
that distribution over time. These findings strongly support the theory, because
there is no other obvious explanation for these differences in difficulty.
The second principle also addresses the gap between subjective and objec-
tive difficulty. Why do college students stare for minutes on end at a problem
that only requires a short sequence of actions that they are entirely capable of
carrying out? Trina Kershaw and I explored this question in the context of the
Nine-Dot Problem: The goal is to draw four straight lines that go through all
nine dots, without backtracking or lifting the pen from the paper. Figure 4.7
53
shows the problem and its solution. Kershaw proposed the Principle of Multiple
Difficulties, which says that the subjective difficulty of an objectively simply
puzzle is caused by the combined action of multiple cognitive constraints, each
representing a small representational change that is required before the most
productive option comes to mind.
Specifically, we hypothesized that the Nine-Dot Problem, though simple,
contains three distinct sources of difficulty: that lines should be drawn outside
the square; that lines should turn in a place where there is no dot; and that the
solution requires diagonal lines. In each case, we could identify prior experi-
ences that would bias thinking away from the useful option. If this is the right
analysis, then providing help on one of these aspects of the problem should
help a little, but only a little. To boost the solution rate in a group of subjects to
the vicinity of 100%, it is necessary to provide help on all three aspects. These
predictions were supported in a series of experiments.
Although the consistency between theory and these experiments is pos-
itive, the experiments constitute only a weak test of the theory. The expec-
tations that shaped these experiments are consistent with common sense.