Janine's Conjecture
2 out of 3
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. Does this always work? Can you prove or disprove this conjecture?
Problem
Janine noticed something interesting while studying some cube numbers. She said, "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."
For example, 3, 4, 5 gives 3 × 4 × 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.