Multiple Proofs 13

of three consecutive numbers, one of which must be a multiple of 3, and so the product is

indeed a multiple of 3.

How about multiples of 4? The high-school proof again generalizes to show that the

product of any four consecutive numbers is a multiple of 4 (and in fact these products are all

multiples of 24, which the reader will prove in exercise 2.7). But meanwhile, 2

4

−2 equals

14, which is not a multiple of 4. So the analogues of the two statements are no longer

equivalent in the case where we consider multiples of 4. This is now a very interesting

situation, since we are led to consider the question: for which numbers k is n

k

− n always

a multiple of k? This question leads into the deeper waters of number theory.

Meanwhile, the geometric proof suggests the idea of going to three dimensions. For

example, we can see that n

3

− n

2

is always even, since you can cut a cube, like a cake, by

laying the knife along the diagonal at the top and cutting straight down, removing the n

2

many cubelets appearing on that slice. What is left has two equal halves, and so n

3

− n

2

is

even. Of course, we could also have observed simply that n

3

− n

2

= n(n

2

− n), which is

therefore a multiple of an even number and hence even.

Mathematical Habits

Try it yourself first. When encountering a new theorem while reading, close the

book or article and try first to prove it yourself. Push your argument through as far

as you can. When you get stuck, consult the original proof for a clue about a concept

or perspective you had missed, and armed with that clue, close the book or article

and try once again to prove the theorem on your own. This way of reading will give

you a greater understanding of the challenges and a greater appreciation for the value

of a key step or idea, which will also enable you to learn the argument very deeply.

Students who read proofs this way will inevitably become strong mathematicians.

Seek multiple proofs. Strive to prove your statements with arguments arising from

totally different perspectives on a problem, which might give insight on different as-

pects of the situation.

Prove a special case. Faced with a difficult or stubborn proof, make an additional

assumption and prove the statement in that special case.