Square Business

Any complete covering of a larger square by 3 unit squares is going to partition the perimeter of the larger square into at least 3 segments. Suppose one of them covers one segment. The maximum segment any one square can cover has a length of $2$ , which occurs when one of the unit squares shares one same vertex with the larger square and is aligned with it---as the graphic shows. Then it's a matter of how much more the other two unit squares can cover.

If $S$ is the side length of the larger square, and point $P$ has the coordinates $(x,y)$ , then if we let $s=\dfrac{y}{x}$ , point $Q$ has the coordinates

$Q= \left( \dfrac{S}{ {s}^{2}+1 }, \; s \dfrac{S}{ {s}^{2}+1 } \right)$

then we have the following system of equations to solve for $x, y, s, S$

${x}^{2} + {x}^{2}=1$
$s=\dfrac{y}{x}$
$-\dfrac{1}{s} (S-x)+y=S-1$
${\left( S-\dfrac{S}{{s}^{s}+1}\right) }^{2}+{ \left( s\dfrac{S}{{s}^{s}+1} \right) }^{2}=1$

which yields the result

$S=\sqrt{ \dfrac{ 1+\sqrt{5} } {2} }=1.27202...$