$$4*(1+3+5+7)=8^2.$$
$4*$(sum of odds from first to $k$th odd number)=area of a square region=$4*k^2$.
$k-1$ is the number of $L$'s.
You can solve the $10\times 12$ problem via considering $10\times 10$ and the extra $1\times 2$ not added via an $L$, one $L$ would be too big. to make the extra is too big.
$14400/4$=total area/area each$=3600,\quad n=60$.
means you use not the 2x2 block as the building block but rather than 64 in^2 block as the building block, a interesting thing is the fnd how many of the 1+3*(4) inch^2 block, how many of the 1+3+5 block, etc.