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The goal is to tile rectangles as small as possible with the given heptomino, in this case number 7 of the 108 heptominoes (see example below). 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 heptomino will tile.
Example with the $1\times 1$ you can tile a $2\times 5$ as follows:
Now we don't need to consider $1\times 1$ further as we have found the smallest rectangle tilable with copies of the heptomino plus copies of $1\times 1$.
I found only 7 more. I considered component rectangles of width 1 through 11 and length to 31 but my search may not be complete.
List of known sizes:
- Width 1: Lengths 1 to 5
- Width 2: Lengths 2, 3, 5
These could all be tiled by hand, of course the bigger ones will be challenging. I'm making this one a 'hand tiling only' puzzle. In other words, use a computer to do anything except look up or compute the arrangements.