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Let us use this approach on the problem of determining a formula which will give us the number of diagonals of a convex polygon in terms of the number of sides. The three-sided polygon, the triangle, has no diagonals; the four-sided polygon has two. An examination of other cases yields the data included in the following table:

n = number of sides	3	4	5	6	7	8	9	...	n	...
D_n = number of diagonals	0	2	5	9	14	20	27	...	D_n	...

The task of guessing the formula, if a formula exists, is not necessarily an easy one, and there is no sure approach to this part of the over-all problem. However, if one is perspicacious, one observes the following pattern:

This leads us to conjecture that

2D_n= n(n - 3) or D_n=

Now we shall use mathematical induction to prove this formula. We shall use as a starting point n = 3, since for n less than 3 no polygon exists. It is clear from the data that the formula holds for the case n = 3. Now we assume that a k-sided polygon has k(k - 3)/2 diagonals. If we can conclude from this that a (k + 1)-sided polygon has (k + 1)[(k + 1) - 3]/2 = (k + 1)(k - 2)/2 diagonals, we will have proved that the formula holds for all positive integers greater than or equal to 3.

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Saturday, April 11, 1998