Mind, Body, World- Foundations of Cognitive Science, 2013a

used to solve a complex problem by reducing it to a simple version of itself. This
problem-solving approach is often called divide and conquer (Knuth, 1997).
One example of this is the famous Tower of Hanoi problem (see Figure 3-1),
first presented to the world as a wooden puzzle by French mathematician Edouard
Lucas in 1883. In this puzzle, there are three locations, A, B, and C. At the start
of this problem there is a set of differently sized wooden discs stacked upon one
another at location A. Let us number these discs 0, 1, 2, and so on, where the number
assigned to a disc indicates its size. The goal for the problem is to move this entire
stack to location C, under two restrictions: first, only one disc can be moved at a
time; second, a larger disc can never be placed upon a smaller disc.
0
1
2
3
4
A
(Start)
C
(Goal)
B
(Spare)
Figure 3-1. The starting configuration for a five-disc version of the Tower of Hanoi problem.
The simplest version of the Tower of Hanoi problem starts with only disc 0 at location
A. Its solution is completely straightforward: disc 0 is moved directly to location
C, and the problem is solved. The problem is only slightly more complicated if
it starts with two discs stacked on location A. First, disc 0 is moved to location B.
Second, disc 1 is moved to location C. Third, disc 0 is moved from A to C, stacked on
top of disc 1, and the problem has been solved.
What about a Tower of Hanoi problem that begins with three discs? To solve
this more complicated problem, we can first define a simpler subproblem: stacking
discs 0 and 1 on location B. This is accomplished by doing the actions defined
in the preceding paragraph, with the exception that the goal location is B for the
subproblem. Once this subtask is accomplished, disc 2 can be moved directly to the
final goal, location C. Now, we solve the problem by moving discs 0 and 1, which are
stacked on B, to location C, by again using a procedure like the one described in the
preceding paragraph.
This account of solving a more complex version of the Tower of Hanoi problem
points to the recursive nature of divide and conquer: we solve the bigger problem by
62 Chapter 3