A grid of dots, and one question: how many triangles have three of the dots as corners? A 3-by-3 grid looks small, but the answer is 76 — and the classic wrong answer, 78, comes from forgetting just two of the straight lines.
The answer key prints on a separate page.
A triangle is a CHOICE of three dots, so the first count is every way to pick three: C(N,3), where N is the number of dots. For a 3-by-3 grid that is C(9,3) = 84. The catch is that three dots on one straight line make no triangle at all, so those triples come back off — and a grid has more straight lines than people remember: three rows, three columns, and the two diagonals, eight lines of three in all. 84 − 8 = 76. Whoever answers 78 has counted the rows and columns and forgotten the diagonals.
The move that makes it tractable is the same one behind counting handshakes and lattice paths: a shape is a choice, and choices are counted without ever being listed. What changes here is the second step — the correction — which is exactly the part a child has to look for rather than compute. A wider grid has straight lines at other slopes too (a line of slope one-half through three dots is still a line), which is why the sheet keeps the grids small enough to check by eye and large enough that guessing fails.