An expert question on counting shapes for P5–P6, with a step-by-step solution, diagrams and the answer. 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.
The question
Triangle ABC has 3 points on side AB, 5 on side BC and 3 on side CA, 11 points in all, as shown.
None of them is a corner:
A, B and C themselves don't count.
Any four points with no three on one straight line join up into a four-sided shape, and each set of four counts as one shape.
How many different four-sided shapes have four of these 11 points as their corners?
Show the hints (3)
💡Hint 1: First count the ways to pick four of the 11 points, then take off the picks that make no four-sided shape. Which ones are they?
💡Hint 2: Three or four points on one side make no four-sided shape. Count each side on its own: all four on that side, and exactly three on it.
💡Hint 3: Say AB has 3 points: exactly three on AB means three of those, plus one of the other 8. Then ask: could one set of four be counted on two sides?
Worked solution
Have a go first, then read the solution step by step.
1.Four of 11 points (the shaded shape is one pick): 11 × 10 × 9 × 8 ÷ 24 = 330 ways.
2.AB has 3 points: three on AB is 1 way, with any of the other 8 points: 8 (like the shaded one: three in a line, so the four make only a triangle).
3.BC has 5 points: all four on BC is 5 ways; exactly three on BC is 10 ways, each with any of the other 6 points: 10 × 6 = 60 (like the shaded one: three in a line, so the four make only a triangle); 65 in all.
4.CA has 3 points: three on CA is 1 way, with any of the other 8 points: 8 (like the shaded one: three in a line, so the four make only a triangle).
5.No set of four has three points on one side and three on another: the sides share no points here, and 3 + 3 = 6 is more than 4. So nothing is taken off twice.
6.330 − 81 = 249 four-sided shapes in all.
⚠️Common mistake: taking off only the picks with all four on one side and missing those with exactly three. Three in a line leave only a triangle.
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