Given an integer n>1 and a n×n grid ABCD containing n2 unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called symmetry if each unit square has center on diagonal AC is colored by gray and every couple of unit squares which are symmetry by AC should be both colred by black or white. In each gray square, they label a number 0, in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called k-balance (with k∈Z+) if it satisfies the following requirements:i) Each pair of unit squares which are symmetry by AC are labelled with the same integer from the closed interval [−k,k]ii) If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct.a) For n=5, find the minimum value of k such that there is a k-balance label for the following grid[asy]
size(4cm);
pair o = (0,0); pair y = (0,5); pair z = (5,5); pair t = (5,0); dot("A", y, dir(180)); dot("B", z); dot("C", t); dot("D", o, dir(180));
fill((0,5)--(1,5)--(1,4)--(0,4)--cycle,gray);
fill((1,4)--(2,4)--(2,3)--(1,3)--cycle,gray);
fill((2,3)--(3,3)--(3,2)--(2,2)--cycle,gray);
fill((3,2)--(4,2)--(4,1)--(3,1)--cycle,gray);
fill((4,1)--(5,1)--(5,0)--(4,0)--cycle,gray);
fill((0,3)--(1,3)--(1,1)--(0,1)--cycle,black);
fill((2,5)--(4,5)--(4,4)--(2,4)--cycle,black);
fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black);
fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black);
fill((4,3)--(5,3)--(5,2)--(4,2)--cycle,black);
for (int i=0; i<=5; ++i) { draw((0,i)--(5,i)^^(i,0)--(i,5)); }
[/asy] b) Let n=2017. Find the least value of k such that there is always a k-balance label for a symmetry coloring. combinatoricsgraph theorysymmetry