Question 162
Let $a$ be a positive integer. Show the following assertion is valid for sufficiently large integer $n$:
From a grid of squares extending to $\pm\infty$ in both $x$- and $y$-directions, choose $n$ unit (i.e., $1 \times 1$) squares and color them black. Let $K$ be the number of $a \times a$ square blocks which contain exactly $a$ black unit squares. Then the maximum possible value $K$ can take is $a(n+1-a)$.
Here, by the statement "An assertion is valid for sufficiently large $n$", we mean that there exists a positive integer $N$ for which the assertion is valid for all $n \ge N$.
From a grid of squares extending to $\pm\infty$ in both $x$- and $y$-directions, choose $n$ unit (i.e., $1 \times 1$) squares and color them black. Let $K$ be the number of $a \times a$ square blocks which contain exactly $a$ black unit squares. Then the maximum possible value $K$ can take is $a(n+1-a)$.
Here, by the statement "An assertion is valid for sufficiently large $n$", we mean that there exists a positive integer $N$ for which the assertion is valid for all $n \ge N$.