Thread: Pairs and pieces

Find all such pairs of positive integers (m,n), for which a rectangle with dimensions of m x n can be built from such pieces created from 6 squares marked as "O" (note that you can both rotate and flip the pieces):

Code:

O O
OOOO

(if it's not clear, it's a F-type shape)

OK. So we now that we can create a 3x4 rectangle by simply putting together this shape and the one rotated by 180^:

Code:

OOOO
 O O

As we can create 3x4, we can also create 4a x 3a for a positive integer a. But what else? I'm kind of clueless...

Hello, GGPaltrow!

Find all such pairs of positive integers ,
for which a rectangle with dimensions can be built
from pieces created from 6 squares as shown.
(Note that you can both rotate and flip the pieces):

. .


OK, so we know that we can create a 3x4 rectangle
. . by simply putting together this shape and the one rotated by

. .


We can also create rectangles for positive integers and
But what else? .I'm kind of clueless ...

Me too . . .

I've tried to find other possible rectangles, but have failed so far.

Hmm.. maybe sth like this:

Suppose we have:

Code:

O O
OOOO

in which the four squares form an edge. Then, let's try to put sth beside it (as putting above leads to a 3x4 rectangle):

Code:

   OO
    O
O OOO
OOOOO

This can't be transformed into a rectangle as our "F" can't be put inside it in any way. The same goes with this flipped:

Code:

    OO
    O
O O OO
OOOOO

Which is even more wrong as we leave our black F laying and blocked from the "open" side so we can only complete it to a 4x3.

Making any F lie on its "back" beside the black one doesn't make sense cause it leads to completing each of them to a 4x3 which means we eveantully end with 4ax3b.

What about this?

Is this the correct solution? Or it needs other formal handling? Or maybe it's wrong?