# smallest square containing k non-overlapping equal rectangles at any orientation

This seems like something that should have a known answer, but I haven't found it after some time alternating between searching and generating multiple pages of algebra. I'm interested in $k=4$ and $k=6$, but I'll phrase it for general $k$:

Given $k$ identical rectangles of dimension $a \times b$ (let's assume $a \leq b$), what is the smallest square that can contain them if the rectangles do not overlap and can be placed at any orientation (i.e., not necessarily parallel to the square's sides)?

• It is not clear from your wording what constraints are given and what freedoms are allowed. For example, for k=1 and a much smaller than b, I can use a square with side length close to 0.7 times b, if I get to choose the packing. Do I get to choose? Or are the orientations fixed and I have to find the smallest square covering them? Gerhard "What Are The Rules Here?" Paseman, 2018.08.31. – Gerhard Paseman Aug 31 '18 at 22:50
• What motivates this question? And why are the cases $k=4$ and $k=6$ of special interest? – Wlodek Kuperberg Nov 6 '18 at 21:50