2014/03/18

Polyominoes consist of edge connected squares. The sides of these squares that are at the perimeter can be notched at their center. A puzzle of 16 V-trominoes with two notches at different points was inventented by Junichi Yananose and you are asked to cover a 6x8 rectangle, where all notches are aligned. Jared McComb wanted to generalize this puzzle and asked for the set of V-trominoes with 0 to 8 notches at the line segments of the perimeter. If you allow to flip the pieces you get 136 pieces and a lot of figures can be constructed. One example is shown in the title.

Other polyominoes can also be used and I explored some sets with a fixed number of notches or a number of notches within a given interval. The main objectives are to fill a rectangle, to align the notches and to get straight borders without notches. Some constraints should be checked before a search for a special construction is started.

- Sometimes it's impossible to fill a rectangle with the pieces even if the notches are ignored.
- If the number of all notches is odd, all notches can't match.
- There must be enough straight edges for the border.
- The number of notches on white or black squares under checkerboard coloring must be equal.

Click on the different polyominoes to see some information about the sets with these pieces. Some constructions are shown and some proofs for impossible cases are given. Since S-tetrominoes can't fill a rectangle combined sets with S-tetrominoes are also explored.

V-Tromino |
L-Tetromino |
T-Tetromino |
Square Tetromino |

I-Tetromino |
S-Tetrominoes plus |
P-Pentomino |
Rectangular Hexomino |

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