If the patio is a square of side $x$, the flower bed is a square of side $y$ and the number of tiles in the border is a square number, then we need to find solutions of the equation $x^2 - y^2 = z^2$. In addition the width of the border has to be a whole number, that is $(x - y)$ has to be an even number.

We now show that $z$ must be an even number. As $(x - y )$ is even it follows that $(x - y)(x + y) = x^2 - y^2 = z^2$ is an even number. Because $z^2$ is an even number we know $z$ cannot be odd (as the square of an odd number is an odd number) so $z$ must be even.

There are an infinite number of solutions to this problems and some of these are:

The solution sets of three numbers are Pythagorean triples. We get solutions by taking $x = k(p^2 + q^2 )$, $y = k(p^2 - q^2 )$ and $z = 2kpq$ where $k$, $p$ and $q$ are whole numbers. The width of the border is then $kq$ . If $k$ is an even number the same Pythagorean triple will give a second solution by exchanging the values of $y$ and $z$ (e.g. $10$, $6$, $8$ and $10$, $8$, $6$).