Pairs of Heptominoes in Rectangles
There is a large number of heptominoes which are not rectifiable and there are numerous combinations of pairs which might form rectangles. No attempt at any detailed analysis has been done but examples are given below where the number at the left denotes the width of the rectangles. (An M in a diagram shows that the solution is due to Mike Reid.)
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Mike Reid has pointed out that the following pairs of heptominoes will also form rectangles in the same way as the 14omino which they combine to make.
There are almost certainly smaller rectangles for each of these pairs all of which are shown above (those supplied by Mike have an M in the bottom right corner). The first pair of the four on the right above can form a 3fold copy of two heptominoes and so can make a large number of other rectangles based on these pieces the smallest of which is a 6x21 rectangle (using two copies of the figure at the right) which, according to Mike Reid, is the smallest for this pair. He has also found a 42x84 rectangle for the third pair on the right.
The six heptominoes below are the only ones (with the exception of the one with a hole) not known to combine with another nonrectifiable heptomino to form a rectangle.
One interesting question is whether a 7x7 square exists for a pair of heptominoes. It is possible if we use two rectifiable pieces or one rectifiable and one nonrectifiable piece (see below).
Mike Reid has found a square using three heptominoes (four other examples based on Mike's solution are also shown) and Patrick Hamlyn has given four possibilities with four pieces.
Patrick Hamlyn has recently produced a set of 244 squares made with three nonrectifiable heptominoes. The set is certainly complete but there may be some duplications. He also showed that there is no pairing of nonrectifiable heptominoes which form the square.
Patrick has also produced a full list of 26 pairs (one rectifiable and the other not) of heptominoes in a 7x7 square.