Checkerboard Problem

Question

Let $a$ and $b$ be any positive integers, and consider an $a\times b$ checkerboard. Let $S(a,b)$ be the total number of different squares of any size on our $a\times b$ checkerboard. What are the values of $n\in\mathbb{N}$, if any, for which $$0.399<\frac{S(n,n)}{S(n,2n)}<0.401?$$ Help would be much appreciated. Thanks in advance.

Answer 1

For both $n\times n$ and $n\times2n$, the squares can have side length $1\le k\le n$. On the $n\times n$ board, there are $(n-k+1)^2$ squares of side length $k$; on the $n\times2n$ board there are $(n-k+1)(2n-k+1)$. Thus we have

$$ S(n,n)=\sum_{k=1}^n(n-k+1)^2=\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}6 $$

and

$$ S(n,2n)=\sum_{k=1}^n(n-k+1)(2n-k+1)=\sum_{j=1}^nj(n+j)=\frac{n(n+1)(2n+1)}6+\frac{n^2(n+1)}2=\frac{n(n+1)(5n+1)}6\;. $$

The ratio is

$$ \frac{2n+1}{5n+1}\;, $$

and this is less than $0.401$ for $n\ge120$.