Olympiad topic · P2–P6
Counting shapes
Every year of the Hua Xia Cup, P1 to P6, has shape-counting questions. The hard part is counting each shape once and missing none.
Try one
★★ MediumInside a large triangle, 2 lines are drawn from the top corner to the base, and 1 line across, parallel to the base, as shown.
How many four-sided shapes are there in the picture?
- Hint 1: In every four-sided shape here, the top and bottom lie on across lines (the base counts), and the left and right on lines from the top corner.
- Hint 2: Count the across lines (base included) and pick two for the top and bottom; then count the lines leaving the top and pick two for the left and right.
- 1.There are only 2 across lines (the line and the base): 1 way to pick the top and bottom.
- 2.4 lines leave the top (the two sides and 2 more); pick two: 3 + 2 + 1 = 6 ways.
- 3.Each top-and-bottom pick with each left-and-right pick makes one four-sided shape (the shaded one is one of them): 1 × 6 = 6 in all.
- ⚠️Common mistake: counting only the small pieces and missing the big shapes spanning several, or taking the triangles at the top for four-sided shapes.
Answer: 6 shapes
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsSegments, squares in a grid.
- ★★MediumFree2 question types · unlimited questionsRectangles containing a dot, quadrilaterals in a triangle.
- ★★★HardFree3 question types · unlimited questionsA grid with a square missing, a triangle cut by many lines.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsQuadrilaterals from points, rectangles avoiding two dots.
Counting shapes: what it is and how to solve it
Counting the segments, triangles, squares or rectangles in a figure.
For example: A line with 5 points on it, the two ends included: how many segments? Counting rightwards from each point: 4 + 3 + 2 + 1 = 10.
The key idea: Count in an order so nothing is missed or counted twice: by size, or by starting point. n points give 1 + 2 + … + (n − 1) segments; rectangles in a grid = ways to pick two horizontal lines × ways to pick two vertical lines.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download