2021/10/05

Given regular octagons with colored edges you can try to arrange the pieces using an orthogonal grid so that touching edges are same colored. This is rather easy. But what about the square holes in such a construction? Jared McComb suggested that opposite sides of the square holes should also be same colored. This way it's much harder to get a valid solution. Rectangular figures have 3 matching conditions at the corners, 5 matching conditions at the border and 8 matching conditions for pieces inside. No wonder that rectangles with long borders are easier to construct than almost squares.

If n colors are available the number of different one-sided pieces is (n^8+4n+2n^2+n^4)/8. For n>2 these numbers are rather high and I decided to look at subsets where only two colors out of n are used for each piece. It's also possible to include or exclude same colored pieces. If three or four colors out of n are allowed you can demand a fixed ratio of frequencies for these colors.

The picture above shows a construction choosing one or two colors out of three for the pieces. Typically pieces with many sides of same color are making kind of clusters.

The table shows the number of pieces for a lot of sets and subsets. Click the numbers to see some constructions.

Colors | One or Two-sided | Number of Colors on Each Piece | Ratio of Color Frequencies | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

≥ 1 | > 1 | ≤ 2 | = 2 | 6:2 | 5:3 | 4:4 | 2:2:4 | 2:3:3 | 1:3:4 | 2:2:2:2 | 1:2:2:3 | ||

2 | One-sided | 36 | 34 | - | - | 8 | 14 | 10 | - | - | - | - | - |

Two-sided | 30 | 28 | - | - | 10 | 8 | - | - | - | - | - | ||

3 | One-sided | 834 | 831 | 105 | 102 | 24 | 42 | 30 | 162 | 210 | 210 | - | - |

Two-sided | 498 | 495 | 87 | 84 | 30 | 24 | 99 | 114 | 114 | - | - | ||

4 | One-sided | 8230 | 8226 | 208 | 204 | 48 | 84 | 60 | 648 | 840 | 840 | 318 | 2520 |

Two-sided | 4435 | 4431 | 172 | 168 | 60 | 48 | 396 | 456 | 456 | 171 | 1296 |