2015/11/17

Take a one-sided V-tromino with three squares. Using three colors there are 3^3 = 27 possible colorings of the squares if repetition of colors is allowed. Jared McComb asked for a Sudoku like 9x9 square made of theses pieces, where the number of same colored squares in each column, each row and each of the nine 3x3 subsquares is always 3. Computer solutions are immediately found and one solution is shown above. The pieces are made of hard foam with magnetic foil on the back to fix the pieces on an iron board.

What is the minimum number of preset pieces to get unique solutions?

The problem of Sudoku like squares can be generalized looking at other polyominoes. Since a one-sided I-tetromino is symmetric under 180 degree rotation there are only 1/2*(3^3+3^2) = 18 different colorings. With a total of 18*3 = 54 small squares you can't get a square. Therefore it seems to be better to take I-tetrominoes with fixed orientation getting 27 different pieces and a Sudoku like 9x9 square is possible.

With square colored tetrominoes you can also construct such squares. All pieces are one-sided and the O- and I-tetrominoes have a fixed orientation. The S-tetromino can't fill a square at all. For two or four colors the following tabel shows the results.

Tetromino Type | Colors | Pieces | Square Size | Same Colored Squares in Each Row | Same Colored Squares in Each Row | Subsquare Size | Same Colored Squares in Each Subsquare |
---|---|---|---|---|---|---|---|

O-tetromino with Fixed Orientation | 2 | 16 | 8x8 | 4 | 4 | 4x4 | 8 |

I-tetromino with Fixed Orientation | 2 | 16 | 8x8 | 4 | 4 | 4x4 | 8 |

T-tetromino | 2 | 16 | 8x8 | 4 | 4 | 4x4 | 8 |

L-tetromino | 2 | 16 | 8x8 | 4 | 4 | 2x2 | 2 |

O-tetromino with Fixed Orientation | 4 | 256 | 32x32 | 8 | 8 | 4x4 | 4 |

I-tetromino with Fixed Orientation | 4 | 256 | 32x32 | 8 | 8 | 4x4 | 4 |

T-tetromino | 4 | 256 | 32x32 | 8 | 8 | 4x4 | 4 |

L-tetromino | 4 | 256 | 32x32 | 8 | 8 | 4x4 | 4 |

Constructions with subsquares of size 4x4 are possible and each of th four subsqaures contains 8 squares of each color.

For the L-tetromino a construction even with 2x2 subsquares can be made.

After the problems were solved I decided to make one set of 256 L-trominoes. The solution is shown on a 75cm x 75cm iron board. Even with the computer solution it took me some time to finish the construction, because it's difficult to spot the correct pieces in the large set. It's a good idea to divide the whole set into subsets due to the frequencies of the different colors .

Home