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.)
Let me first say that this is one of those Fiddlers that I wish I had not programmed up to try to verify my scratch work. I had a bunch of lovely handwritten work that mistakenly classified the only possible equilateral triangles on the Dozo board as those that had up, down, left or right orientation, with respect to an $xy$-plane that chose one side of the triangular board as the $x$-axis. Examples of up-, down-, left-, and right-oriented equilateral triangles are shown below in red, blue, orange and green, respectively. If I had stuck with the combinatorial work I had done with these up, down, left and right oriented equilateral triangles, then the ultimate answer would have be .... 100, oh the delightful, thoughtful deliberate work to get an answer that aligns with the official tally of how many of these we have ever done.
But alas, to confirm, I outsourced the work to some nice computer by defining the coordinates of the holes on the Dozo board and then running through all triplets of holes to find cases where the distances between each vertex are identical. Let's first define the coordinates. Let the top of the Dozo board, as shown, be the $x$-axis, with the seven holes located at $(-3,0)$, $(-2, 0)$, $\dots$, $(2,0)$, $(3,0).$ This means that the distance between each vertex with have unit length, and means that the second row should then be $(-2.5, -\frac{\sqrt{3}}{2})$, $(-1.5, -\frac{\sqrt{3}}{2})$, $\dots$, $(1.5, -\frac{\sqrt{3}}{2})$, $(2.5, -\frac{\sqrt{3}}{2}),$ and so on and so forth until we get to the final lowest vertex at $(0, -3\sqrt{3}).$
As described above, once all $28$ of the vertices were coded up, the search for all possible equilateral triangles loops through all possible unique triplets $(i, j, k) \in \{1,2,\dots,28\}^3$ with $i \lt j \lt k$ and tests whether $$\|v_i - v_j\|_2 = \|v_j-v_k\|_2 = \|v_i - v_k\|_2.$$ In this case, it yields all 100 of the up, down, left and right oriented equilateral triangles along with 26 additional ones for a total of $126$ equilateral triangles on the Dozo board. Some examples of the equilateral triangles that do not have one of the four orientations given above are shown in the chart below, with the fuschia triangle having side lengths of $\sqrt{7}$, the yellow triangle having side lenghts of $\sqrt{13}$, and the purple triangle having side lengths of $\sqrt{21}.$ There are a total 18 possible equialteral triangles with side lengths of $\sqrt{7}$, 6 with side lengths of $\sqrt{13}$ and 2 with side lenghts $\sqrt{21}$.
