`Proof that a rectangle can't be tiled by XX XX:`
`One polyomino must end in a corner of the rectangle. So we have:``
`

`+--------`
`|AA AA`
`|``
`

`The gap can only be filled by a polyomino crossing in the other`
`direction.``
`

`+--------`
`|AABAA`
`| B`
`|`
`| B`
`| B``
`

`Now, the gap in B can only be filled in two ways. Both require
a`
`polyomino`
`crossing parallel to A, but in different positions. First
consider:``
`

`+--------`
`|AABAA`
`| B`
`| CC CC`
`| B`
`| B``
`

`Now, the gap between the second square of A and the first of C can
only`
`be filled by a polyomino directly under and parallel to A.
This also`
`leaves only one way to fill the gap to the left of C:``
`

`+--------`
`|AABAA`
`|DDBDD`
`|ECC CC`
`|E B`
`| B`
`|E`
`|E``
`

`Now, again there is only one way to fill the gap just below the
first`
`square in C. But adding this polyomino makes it impossible
to fill the`
`gap in E:``
`

`+--------`
`|AABAA`
`|DDBDD`
`|ECC CC`
`|EFB`
`|*FB * = impossible to fill`
`|E`
`|EF`
`| F``
`

`Now, consider the other position for C:`
`+--------`
`|AABAA`
`| B`
`| CC CC`
`| B`
`| B``
`

`The gap between A and C can only be filled by a polyomino parallel
to`
`them,`
`but there are two possible places for it. One pair of the
x's in the`
`follwing figures must contain the other two D's:``
`

`+--------`
`|AABAA`
`|xxBDD xx`
`| CC CC`
`| B`
`| B``
`

`Now, the gap in C can only be filled one way, and the gap thus created`
`between B and C can only be filled by another polyomino parallel
to`
`them:``
`

`+--------`
`|AABAA`
`|xxBDD xx`
`| CCECC`
`| BFE`
`| BF`
`| E`
`| FE`
`| F``
`

`Now, the problem here is on the left edge. If D extends to
this edge,`
`then`
`the spaces below it can only be filled by placing two polyominos`
`vertically,`
`but this leaves a two-square gap at the left end of the fifth row
which`
`cannot be filled:``
`

`+--------`
`|AABAA`
`|DDBDD`
`|GHCCECC`
`|GHBFE`
`|**BF`
`|GH E`
`|GH FE`
`| F``
`

`And otherwise, the same thing happens but the gap is in a different`
`place:`
`+--------`
`|AABAA`
`|GHBDD DD`
`|GHCCECC`
`|**BFE`
`|GHBF`
`|GH E`
`| FE`
`| F`
`
`