## One-sided Heptominoes

There are 196 one-sided heptominoes and I sawed these pieces from parquet wood. The edges were black marked and at the back I fixed little magnets to display my constructions on iron board. Because of the piece with an hole you can't get a rectangle figure with these pieces and even if a single hole was allowed the total area would be 196*7+1=1373, which is a prime. Therefore a rectangle can't be constructed either. But the jagged 23x31 rectangle which is shown below is a nice alternative. You can also get a jagged 8x91 rectangle .

If we allow additional holes we can make some rectangles with straight borders. For example a 32x43 rectangle with four holes is shown below. Rectangles of size 6x229 with two holes, 11x125 with three holes and 9x153 with 5 holes are also possible.

What about squares? Starting with a 38x38 square all pieces can only cover 1372 places leaving 72 holes. I arranged these holes along the diagonals getting four congruent triangles connected by the four uniquely determined pieces, that can be placed at the corners.

An empty square of size 8x8 at the center together with 8 additional holes makes a kind of ring.

You can also place the 72 holes at equidistant positions of a diagonal grid.

Starting with a larger square you can get a double ring with four axis of symmetry, but the two rings don't match the same grid.

Multiple constructions are possible, too. You can construct 14 rectangles of size 9x11 with a center hole.

On Mathpuzzle you can see 14 squares of size 10x10 with two holes on a diagonal. In my construction the holes are one unit closer to the corners.

Jagged squares with a center hole are interesting, because for n=7, 8, 14, 15, 21, 22 the total area is a multiple of 7. Fortunately combinations of three jagged squares 7, 14, 22 or 8, 15, 22 can be covered by the complete set of one-sided heptominoes.

Discarding the piece with an hole a lot of retangles can be made as shown on Andrew Clarke's Poly Pages. I tried to construct some similar hole figures of the heptominoes. First I made six 14x14 squares leaving pieces with good tileability. Using these pieces I got the last square with the heptomino hole.

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