Quote:

Find all such pairs of positive integers $\displaystyle (m,n)$,

for which a rectangle with dimensions $\displaystyle m\!\times\!n$ can be built

from pieces created from 6 squares as shown.

(Note that you can both rotate and flip the pieces):

. . $\displaystyle \begin{array}{cc}\square \;\,\square \;\; \\ [-2mm]\square\! \square\! \square\! \square \end{array}$

OK, so we know that we can create a 3x4 rectangle

. . by simply putting together this shape and the one rotated by $\displaystyle 180^o.$

. . $\displaystyle \begin{array}{c}\blacksquare\! \blacksquare\!\blacksquare\! \blacksquare \\ [-2mm] \square \! \blacksquare\! \square\! \blacksquare \\ [-2mm] \square \! \square \! \square \! \square \end{array}$

We can also create $\displaystyle 4a\times 3b$ rectangles for positive integers $\displaystyle a$ and $\displaystyle b.$

But what else? .I'm kind of clueless ...