Sums of squares and then increasing the number being square: will the sum change?

Question

100 numbers are written on a board. After each number was increased by 1, the sum of their squares did not change. Will the sum of squares change and by how much if the numbers are increased by 1 again?
I don't know where to start since I can't find an example and that logically the sum will always change because the squares will always be non-negative.

Answer 1

Hint:   in the case of $\,3\,$ numbers, the premise is: $$(a+1)^2+(b+1)^2+(c+1)^2 - (a^2+b^2+c^2) = 0 \;\;\iff\;\; 2(a+b+c)=-3$$

Then $\,(a+2)^2+(b+2)^2+(c+2)^2 - \big(a^2+b^2+c^2\big) = 4(a+b+c)+12=6\,$.