Octominoes

Unfortunately this set is unbalanced (chessboard colouring) and so no rectangles is possible. Eleven congruent shapes can, however, be made.

Also possible with this set is to make three 11-fold replications of an unbalanced octomino.

Patrick Hamlyn has produced the following construction which, by rotating the various sections, will produce solutions to the above problem for a number of the unbalanced octominoes. Some examples of the other pieces formed are shown below.

There are 227 octominoes which are chessboard balanced (4-4 colouring) and it is tempting to try to form a rectangle with these pieces. Unfortunately this is not possible since if we colour the possible rectangles by strips we find that and odd number of the pieces cover an unbalanced colouring.

for octominoes , and other n-ominoes with n divisible by 4, we have these parities:

12-21 this notation means that we tile the plane with aligned squares with colors 1 and two, the first row is (1,2), the 2nd row is (2,1)
Parities are determined by a total of an even number of pieces of each colour.
1	12-21	230 non-holey 8ominoes have parity1=0 and 133 have parity1=1
2	11-22	this is rotation invariant, iff parity1=0 parity2 of a piece is ( number of covered cells colored with 1 ) modulo 2 124 of the non-holey 8ominoes with parity1=0 have parity2=0.
3	12-34	this is invariant, iff parity1=0=parity2 64 of the 124 pieces above have parity3=0
4	1122-1122-2211-2211	this is invariant, iff parity1=0=parity2 64 of the 124 pieces above have parity4=0 , (not the same pieces as in (3), 25 have parity3!=parity4 )
5	1122-1122-1122-1122
6	1122-1122-3344-3344
7	1234-2341-3412-4123	121 pieces where this is well-defined, 72 pieces where all counts are even, 49 pieces. where all 4 counts are odd
8	1234-3412-1234-3412
9	123-231-312	78 pieces where all 3 counts are even no odd pieces here, the other 152 pieces have non defined parity(9)
10	123-123-123

Patrick Hamlyn has produced the following construction using the 136 unbalanced pieces.