Sets Based on Polydominoes
The following solutions have been found using Peter Esser's solver which can be downloaded from http://members.tripod.de/polyforms. If you find any other interesting solutions using the solver and would like them included here please send them to me. Some diagrams contain figures not yet solved.
The definition of a sliced rectangle should be clear from the diagrams. It is merely a rectangle with a corner sliced off with the type of slice depending on the type of polyform used. Here we concentrate on single slices (one corner only) but multiple sliced could be considered.
Livio Zucca has found the number of solutions for the above figures (some are unique) and has produced some more.
one piece used twice
The full set can also form sliced rectangles.
one-sided set
The only possible rectangles are 5x63, 7x45, 9x35 and 15x21 all of which can be made as well as some similar hole figures. The only sliced rectangles which could be made are a 17x19 with a slice of 4 and a 9x37 with a slice of 6.
one piece used twice
A number of sliced rectangles can be made with this set.
These solutions are by Roel Huismann.
A number of sliced rectangles may be made with this set.
Sliced rectangles are possible with this set.
The only rectangle which might have been possible with this set is a 5x7 but it cannot be made.
This, and similar sets, has two parity considerations. One is the usual colouring parity but the second relies on how the two sloping edges of the pieces occur. They are either parallel or perpendicular. In order for a rectangle to be possible the number of colour unbalanced pieces and the number of perpendicular edge pieces must be even. Unfortunately neither is the case and this shows that not only no rectangle can be made but also no symmetrical construction is possible.
If one piece is used twice then rectangles are possible. With the full set a number of sliced rectangles may be made.
