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 |