# Janine's Conjecture

## Problem

Janine noticed, while studying some cube numbers, that ``if you
take three consecutive whole numbers and multiply them together and
then add the middle number of the three, you get the middle number
cubed''; *e.g.*, 3, 4, 5 gives 3 x 4 x 5 + 4 = 64, which is
a perfect cube. Does this always work? Can you prove or disprove
this conjecture?

## Student Solutions

Julia of Downe House School gave the neatest solution to this problem by substituting 'x-1', 'x', 'x+1' for the three consecutive numbers and giving the following statement of Janine's conjecture:

(

*x*- 1)

*x*(

*x*+ 1) +

*x*=

*x*

^{3}.

This is Julia's proof:

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

and

( *x* ^{2} - 1) *x* = *x* ^{3}
- *x* .

Therefore ( *x* - 1) *x* ( *x* + 1) +
*x* = *x* ^{3} .

So Janine's conjecture will always work whichever three consecutive
numbers are chosen.