Looking at the "

Square of an Odd" puzzle that asks to prove that the square of an odd number is always 1 more than a multiple of 8, a professor gave this four parts proof: "All odd numbers are of the form 8K+1, 8K+3, 8K+5 or 8K+7. Squaring these numbers produces 8M+1, 8M+9, 8M+25 or 8M+49, which are all of the form 8N+1. QED"

Another professor came by, and gave a single line proof. Can you manage it?

*Note: no one who answered the original problem produced either the four parts solution, or the single line one.*

Any odd number, when written in binary, is ...fedcb1, where **b**, **c**, **d**, **e**, and **f** are the multiples of the powers of 2, 4, 8, 16, and 32, respectively. (The number is equal to 1 + 2b + 4c + 8d + 16e + ...)

Squaring this number will yield 1 + 4b + 8c + 4bb + 16d + 8bc + 32e + 16bd + 16cc + ... , which equals 1 + 8k for all {0,1} values of b,c,d,e,f,g,...