Solution
Triangles in a Triangle
Show the problem again
An equilateral triangle of side 6 is divided into 36 small equilateral triangles of side 1 by lines parallel to its sides. How many equilateral triangles of any size (pointing up or down) appear in the figure, with sides along the drawn lines? Give an integer.
Worked solution
The answer is 78. Upward triangles of side $k$: their top vertices can be at any point in the top $6 - k + 1$ rows of the lattice, giving $\binom{6 - k + 2}{2}$ positions, so the count is $\sum_{k=1}^{6}\binom{8-k}{2} = 21 + 15 + 10 + 6 + 3 + 1 = 56$. Downward triangles of side $k$ need room both above and below; they fit inside a band of the big triangle, and there are $\binom{6 - 2k + 2}{2}$ of them for $2k \le 6$: $k = 1$: 15, $k = 2$: 6, $k = 3$: 1, total 22. Overall $56 + 22 = 78$.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.