# Sets of Edge Colored Octagons

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