Question 366789
Let's make a table of n^2 modulo 8 .

0^2 modulo 8 = 0
1^2 modulo 8 = 1
2^2 modulo 8 = 4
3^2 modulo 8 = 1
4^2 modulo 8 = 0
5^2 modulo 8 = 1
6^2 modulo 8 = 4
7^2 modulo 8 = 1

Exhaustive search shows that one cannot find 3 numbers whose sum
is 7 modulo 8.