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The goal is to tile rectangles as small as possible with the given hexomino, in this case number 2 of the 25 hexominoes which cannot tile a rectangle alone. We allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one of the given hexomino will tile.
Example shown, with the $1\times 1$ (or the $1\times 2$) you can tile a $2\times 4$ as follows:
Now we don't need to consider $1\times 1$ (or $1\times 2$) further as we have found the smallest rectangle tilable with copies of the hexomino plus copies of $1\times 1$ (or $1\times 2$).
I have tilings for another nine rectangles: Hexomino plus $1\times 3$, $1\times 4$, $1\times 5$, $1\times 6$, $1\times 7$, $1\times 8$, $2\times 2$, $2\times 3$, $2\times 4$
No Computer All of them could be tiled by hand (with significant effort in some cases), so I'm making this a no-computer puzzle. This also means please don't look up answers on the web... if you post an answer it should be because you found it 'by hand'. This does not preclude you from for example using an image program to manipulate shapes on the screen, just from using a computer to search for or automate the arrangement.