if x^2 is odd then x is odd; let x = 2k +1
then x^2 = (2k+1)(2k+1) = 4k^2 + 4k + 1
1 - (4k^2 + 4k + 1) = -4k^2 - 4k = 2(-2k^2 - 2k) which is of the form 2k which proves it's even
(4k^2 + 4k + 1) + 1 = 4k^2 + 4k + 2 = 2(2k^2 + 2k + 1) which again is of the form 2k which proves it's even.