Janine's Conjecture
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?
Age
14 to 16
Challenge level
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.