This week the #puzzle is: Happy 100th Fiddler! #counting #permutations #triangles #parallelograms #dozo #flag

Dozo is a strategy game with a rather distinctive board:

The board features 28 holes in which players place markers, with the goal of making an equilateral triangle of any size with one color.

How many distinct equilateral triangles can you find whose vertices are the centers of holes on the board? (If two triangles are congruent but have different vertices, they should still be counted as distinct.)

And for extra credit:

Happy Fourth of July! In celebration of America’s birthday, let’s count more shapes—not in a board game, but in the American flag:

In particular, consider the centers of the 50 stars depicted on the flag. How many distinct parallelograms can you find whose vertices are all centers of stars? (If two parallelograms are congruent but have different vertices, they should still be counted as distinct.)

Highlight to reveal (possibly incorrect) solution:

Program Handcount 1 and 2

I tried counting triangles both by hand and with a program. The handcount illustrated is actually a little wrong, because in a certain case (6 + 6) I forgot to count the triangles, where the tip pointed up. The program goes through all the options for the 1st and 2nd vertex, constructing the 3rd and testing whether this hypothetical 3rd is on the board. This was a little tricky, because my model of the board didn’t preserve angles and distances from the board. But I think I got there in the end. Result: 126 triangles.

And for extra credit:

Program

This time I went straight to the program. It goes through all the options for 3 vertices, constructing the 4th. There’s actually a charming set of formulas for constructing the 3 options for the 4th vertex. This time it didn’t matter that angles and distances weren’t preserved. Only tricky part was to detect, when the 1st, 2nd and 3rd vertex were on a line. Result: 5918 parallelograms.

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For the 3rd time I’ve come back to weaving the web. This time because a friend kindly helped me turn a 3d object file into a physical object: