Any product of two evens or two odds (sticking just to positives for the purpose of this problem) can be expressed as a difference of two perfect squares. 11*17=187=196-9 is an example.
A: Prove this idea.
B: Come up with a formula that gives the two perfect squares. Call the larger one a and the smaller one b.
(In reply to Puzzle Solution
by K Sengupta)
SOLUTION TO PART B:
We know that:
((a+b)/2)^2 - ((a-b)/2)^2 = ab
It has been verified in terms of solution to Part A, that the above relationship will hold and the pair ((a+b)/2, (a-b)/2) will always be an integer irrespective of whether (a,b) are both odd or both even.
In addition, when (a,b) are both odd, we obtain the relationship:
ab = ((ab+1)/2)^2 - ((ab-1)/2)^2, where both ((ab+1)/2, (ab-1)/2) are integers.
Edited on November 24, 2007, 10:28 am