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 slices could be considered.
If we use one piece twice then a number of rectangles are possible as well as three or five congruent pieces. The 3x40 rectangle is by Roel Huisman.
With the full set a number of sliced rectangles can be made.
If we use a chessboard colouring on these pieces then we see that 17 have a colouring of 2¼ to 1¾ an excess of ½ whereas the remaining 12 pieces have a colouring of 2¾ to 1¼ which gives an excess of 1½ (see below). This will mean that the piece used twice must be one of the 17 pieces with an excess of ½ using an analysis similar to that for the domsliced tetrominoes.
Monomino + two half squaresAs an example here are the remaining 16 solutions for the 5x24 rectangle above.
Three rectangles can be made with one piece left over.
The figure at the top left is by Roel Huismann
At least one rectangle and two sliced rectangles are possible as well as a similar hole problem.
It is also possible to make a number of sliced rectangles with this set. Notice that one solution can form two different sliced rectangles depending on whether the rectangle is placed at the side of below the sliced section.
With just the full set it is possible to make sliced rectangles.
This set cannot form rectangles or sliced rectangles since any figure formed must have an excess of 4n+2 squares by colouring.
The square at the top left is by Peter Esser. Each figure in the bottom row is made up of two congruent halves.
It is also possible to make a fourfold replica of some of the pieces if we use the replicated piece three times.
These constructions are by Peter Esser. Notice that the lower two are made of two congruent parts.
There is just one sliced rectangle which can be made.
It is also possible, but very difficult, to pack the pieces into three congruent shapes. Three 9x9 squares have been found by Peter Esser
We can also consider a similar set - the tromsliced hexominoes. The is set can be obtained from the above by removing the two pieces where removing half a tromino from an hexomino would take away an internal join between squares. The 6x39 rectangle below was found by Roel Huismann who also found the two squares made from the set.
This set consists of 126 pieces with a total area of 693 squares and can make a number of simultaneous rectangles as shown here (solutions by Peter Esser). No 3x231 rectangle can be made as there are pieces with width four in two directions. A number of parallelograms are also possible (solutions by Peter Esser).
The figure below shows a construction which can make the 7x99 rectangle, parallelogram or trapezium.
It may be possible with this set to remove one piece and then form five fivefold copies of that piece with the rest of the set. The below shows one possibility.
There are also two sliced rectangles which might be possible - 9x83 with a size 6 slice and a 21x47 with a size 14 slice.
This set consists of 172 pieces including on with a hole. Roel Huisman has constructed the 43x22 rectangle below by omitting the piece with a hole and using another piece twice.
A number of sliced rectangles is possible with the set if we omit the piece with a hole.
This set consists of 107 pieces and Roel Huisman has constructed the rectangles below by omitting one of the pieces.
A number of sliced rectangles is be possible with the full set.