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:

(

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.