Hello utopiaNow

It's your proposition that's wrong. You said:andQ: for each , the integer and the sum of any 2 of the remaining 3 integers of S are of different parity.

But you don't need the word any. You need simply show that , such that and are the same parity.: there exists , such that the integer and the sum of any 2 of the remaining 3 integers of S are of the same parity.

Grandad