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Product of Three Successive Integers

A shortcut, maybe, to multiplying three successive integers in your head -- and more.

Having a mathematical bent affords me abundant opportunities for mental amusement. For example, one day, as I was riding my bicycle in my neighborhood, I happened to notice the house-number on one of the mailboxes.

1728.

“Oh, twelve cubed,” I said to myself.

It was a few days later before I noticed the number on the mailbox next door. Ordinarily, around here, adjacent houses have numbers very close together; they may differ by two or four, but rarely much more than that. In this instance, however, the difference was considerably greater.

1716.

“Heh, look at that,” I said to my favorite audience. “Twelve less than twelve cubed.” And my favorite audience -- myself -- told me that it was an instance of . . .

x^{3} - x

And before my bicycle even got me to the next corner, I had factored out an x to get . . .

x (x^{2} - 1)

That parenthetical term is one of my favorites in algebra. It just tickles me no end that . . .

x^{2} - 1 = (x + 1) (x - 1)

Substituting those two first-order terms in place of the second-order term, above, and rearranging them in ascending order of magnitude, I saw immediately that . . .

x^{3} - x = (x - 1) x (x + 1)

This sort of formula can be used for any succession of three numbers at equal intervals
-- 12 x 15 x 18, for example. The exact form of that formula is left as an exercise for the reader.

In other words, the product of three successive positive integers is equal to the middle number cubed minus that middle number. Admittedly, I don't have frequent occasion to multiply three successive numbers, but when I do, I'll have a short-cut handy -- if the middle number is one I can easily cube, that is.