Working Backwards

Entry point

Divide a triangle into three parts in such a way that the parts can be put together again to form a rectangle or a square.

The problem is quite a difficult one since the shape of the triangle is not specified. You first have to choose a triangle shape and then find out how it can be divided up into three pieces that can be put together to give the square or rectangle.

The solution diagrams are not included in this excerpt. It is obviously much easier to start with the square instead of with the triangle which was suggested as the starting point. There can be no doubt about the shape of a square whereas the shape of a triangle (and to a lesser extent of a rectangle) is variable. Since the three parts have to fit together again to form a square one can solve the problem by dividing up a square into three parts that can be put together again to give a rectangle or a triangle. The source illustrates two ways to do this.

In many children’s books there is the sort of puzzle in which are shown three fishermen whose lines have got tangled up. At the bottom of the picture a fish is shown attached to one of the lines. The problem is to find which fisherman has caught the fish. The children are supposed to follow the line down from the tip of the fishing rod in order to find which line has the fish at the end. This may involve one, two or three attempts since the fish may be on any of the three lines. It is obviously much easier to start at the other end and trace the line upwards from the fish to the fisherman. That way there need never be more than one attempt.

There is a simple problem which requires one to draw the outline of a piece of cardboard which is so shaped that with a single straight cut the piece can be divided into four smaller pieces which are exactly alike in size, shape and area. No folding is allowed.

The source’s page 159 shows responses and their percentages; that diagram is not included here. The solution given by groups B and C is obviously incorrect for a ‘cut’ has no thickness and so will divide the shape into two pieces and not four as required.

The source identifies answer D as correct. It is interesting that answer F is so rare for in hindsight it seems the easiest of them all (the explanation is that it is very difficult to think forward asymmetrically and in answer F the pieces are not all used in the same way). The important point of this problem, however, is that if one starts at the wrong end the problem is much easier to solve. Instead of trying to devise a shape that can be divided into four equal pieces one starts off with four equal pieces and clusters them around an imaginary cut. The source’s page 160 illustrates how shifting the pieces along the cut produces a solution; that diagram is also absent here.

To start at the wrong end and work backwards is quite a well-known problem solving technique. The reason why it is effective is that the line of thought may be quite different from what it would have been had one started at the beginning. There is no need to actually start at the solution end. It is convenient to do so since the solution is often clearly defined. But one can start at any point. If there is no obvious point then one must be generated.