Omni-MATH / Let n≥3 be an odd number and suppose that each square in a n×n chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and sh…

Let n≥3 be an odd number and suppose that each square in a n×n chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares a,b are considered connected if there exists a sequence of squares c₁,…,c_(k) with c₁=a,c_(k)=b such that c_(i),c_(i+1) are adjacent for i=1,2,…,k−1. 
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Find the maximal number M such that there exists a coloring admitting M pairwise disconnected squares.

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$. 
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Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.