Solutions To Mensa Algebra Squares Problem

We begin by dividing the positive integers into three groups:

(A) Odd positive integers, 1,3, 5, 7, 9 ....etc.

(B) Even positive integers of the form 4N, e.g. 4, 8, 12, 16...

(C) Even positive integers of the form 4N + 2, e.g. 2, 6, 10, 14, 18...

We note all positive integers (2N +1) CAN be written as a difference of two squares, i.e.

(2N + 1)  =  (N + 1) ² -   N ²

We note all even positive integers  of the form 4N CAN ALSO be written as a difference of two squares, i.e.

(4N) =  (N + 1) ² -   (N  -  1)  ²

But all even positive integers of the form 4N + 2  cannot be written as a difference of two squares.

A difference of two squares is even when:

-  Both squares are even, but all even squares are multiples of four, so their difference is a multiple of four, e.g.

(2A) ²   -   (2B) ²  =  (4A) ²  -   (4B) ²   =  4( A ²   -  B ² )

-   Both squares are odd, but all odd squares are one more than a multiple of four, so their difference is a multiple of four:

(2A + 1) ²   -   (2B + 1) ²  =  4A ²  + 4A + 1 -   4B ²  - 4B - 1

=  4( A ²  + A -  B ² - B)