Figure counting (triangles, squares, rectangles, straight lines) came in ASO 2012 (three), 2018 (four) and 2021 (three). Systematic counting matters more than speed here, because the options are usually close together.
Triangles
- Name every corner and every crossing point with a letter.
- Count the smallest triangles first, then the triangles made of 2 pieces, 3 pieces, and so on.
- Tick each triangle off as you count it, so you don’t count it twice.
- A square with both diagonals has 8 triangles.
- A triangle with k lines from the top vertex to the base and h lines parallel to the base has (k+1)(k+2)/2 × (h+1) triangles.
Squares and rectangles in a grid
- An n × n grid has 1² + 2² + … + n² squares (3 × 3 → 14).
- An m × n grid has C(m+1, 2) × C(n+1, 2) rectangles (2 × 2 → 9).
- A rhombus grid works the same way as a square grid.
Straight lines
Count each full straight line once, even if other lines cut it into pieces.
ASO 2012 figure (square, inner square, both diagonals and both middle lines): 32 triangles, 10 squares and 12 straight lines.