smallest perimeter of compound shape

Question

If you have a compound shape made of three unique squares with fixed sizes, what is the smallest possible perimeter for that shape? assuming no overlaps.

Answer 1

Let the squares $A,B,C$ have respective side lengths $a\le b\le c$.

The total perimeter will be $4a+4b+4c$ less the lengths of any common edges of each pair of $A,B,C$. (If two squares have a length $l$ in common then we must subtract $2l$).

The maximum possible lengths of common edges are $2a,2a,2b$ and so the minimum perimeter is $2b+4c$. This minimum can be achieved if $c\ge a+b$ since we can simply put both $A$ and $B$ on the same edge of $C$.

If, however, $c<a+b$ then the same arrangement as before is best but now the length of the common edges is only $2a+2c$. The perimeter is therefore $2a+4b+2c$.