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.
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.
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. The table below shows the pieces callsified under these two criteria.
The above shows that any figure made with the full set must have a colour excess of an odd number of squares and, since the total area is 72, no rectangle can be made. The only way to get a figure with an excess of an odd number of squares is to have half squares removed and, since the sloping parity is even (i.e. parallel), no symmetrical figure can be made.
If one of the balanced, parallel pieces is used twice then a 5x15 rectangle is possible.
Symmetrical figures are possible with the one-sided set however.
If one of the balanced perpendicular pieces is used twice then rectangles are possible.
one piece used twice
A number of sliced rectangles can be made with this set.