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,...

Posted by Duncan Garp on 2005-06-04 05:34:12