Barge Problem: An intergalactic intervention of geometric proportions

It’s your kid brother Mikey’s birthday and your parents bought him a zillion L-shaped blocks of the form

That evening he is in his room having a great time building all sorts of fun structures and shapes with these blocks. Being the older (meaner) sibling you go into his room and tell him that unless he builds only boring 2-dimensional rectangles with one type of block, the other scary aliens will come down and take him away – forever! He looks at you with big tearful eyes and starts to cry. Having spoiled his fun you head to bed.

Later that night you wake to find a giant tractor beam pulling you from bed to a UFO in the sky. Once on board this ship aliens in the form of really dorky math professors give you the following ultimatum: “Properly determine which rectangles can be created using only type A blocks (respectively, type B blocks) or you will be sent to the furthest reaches of space – forever!”

What will your fate be?