N Squares and One Sliced Square

2022/08/18
Two decades ago Andrew Clarke explored sets of sliced polyominoes, shown on his PolyPages. For some sliced rectangles like the one shown above a construction seemed to be hard or even impossible those days due to the length of the sliced side. Therefore I had a second look at these pieces and possible construction with them.

If you join n squares and one diagonal sliced square, called tan, you get as many different pieces as if you take (n+1)ominoes and remove one tan so that the pieces don't fall apart. These pieces are called n½ ominoes.

If only single cuts are allowed the set is slightly smaller and the pieces don't have interior angles of 315°. (S condition)

Otherwise you can take n-ominoes and extend these pieces by a tan. In this case the set is also slightly smaller because all pieces can't be made this way. (X condition)

Therefore we can look for constructions with the whole set or subsets, where one or both conditions are met. For n=4 the whole set of two-sided pieces is shown. Pieces which violate the S, X or both conditions are marked green, red or yellow, respectively.

Click the numbers to see some construction with the different sets.

Two-sided Pieces One-sided Pieces
Number of Squares Additional Property Number of Pieces Area Number of Pieces Area
3 14 49 27 94.5
X 12 42 23 80.5
S 13 45.5 25 87.5
X & S 11 38.5 21 73.5
4 54 243 106 477
X 44 198 88 396
S 47 211.5 92 414
X & S 38 171 76 342
5 210 1155 417 2293.5
X 171 940.5 339 1864.5
S 175 962.5 347 1908.5
X & S 143 786.5 283 1556.5