Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
15 questions · official format, shuffled order
1 For a triangular prism, what is the number of faces plus the number of edges plus the number of vertices?
Show worked solution
5 faces + 9 edges + 6 vertices = 20.
2 A pyramid has 9 faces. How many edges does it have?
Show worked solution
An octagonal pyramid: 8 base edges + 8 slanting edges = 16.
3 This net folds into a cube. Which face ends up opposite face C?
Show worked solution
Down the column A, C, E, F: A is opposite E and C is opposite F. B and D, either side of C, are the remaining pair.
4 A large cube is built from 4 × 4 × 4 small cubes and every outside face is painted. The cube is then taken apart. How many small cubes have paint on exactly two faces?
Show worked solution
Each of the 12 edges has 4 cubes, of which 2 are corners, leaving 2. 12 × 2 = 24.
5 Here are two views of the same numbered cube. Which number is on the face opposite 3?
Show worked solution
Next to 3 we see 1, 2, 6 and 5 — four neighbours. The only number left is 4, so 4 is opposite 3.
6 This is the view from above of a stack of cubes. The number in each square tells how many cubes are piled there. How many cubes are in the stack altogether?
Show worked solution
2 + 1 + 0 + 3 + 1 + 1 + 1 + 0 + 2 = 11 cubes.
7 Which net folds into a square pyramid?
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A square with a triangle joined to each of its four sides folds up so the four triangles meet at the apex. The triangle-with-three-triangles net makes a triangular pyramid.
8 A prism has 14 vertices. How many edges does it have?
Show worked solution
Each end has 7 edges (14 in total), and 7 edges join the ends: 21.
9 A 3 × 3 × 3 cube is painted on all six faces and then split into 27 small cubes.1 8 small cubes have three painted faces.
2 12 small cubes have exactly one painted face.
3 1 small cube has no paint at all.
Which statements are correct?
2 12 small cubes have exactly one painted face.
3 1 small cube has no paint at all.
Which statements are correct?
Show worked solution
The 8 corners have three painted faces (1 ✓). Exactly one painted face: the centre of each of the 6 big faces — 6, not 12 (2 ✗). The single centre cube is unpainted (3 ✓). 1 and 3.
10 A solid has 7 faces and 10 vertices. How many edges does it have?
Show worked solution
It is a pentagonal prism: 5 + 5 edges on the ends and 5 joining them, 15.
11 Five squares are joined as shown. Which of the dashed squares P, Q or R could be added to make a net that folds into a cube?
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P only. P is a cap on the other side of the band. Q is a second cap on the same side as the existing one, and R makes two caps side by side — both overlap when folded.
12 Which solid has exactly twice as many edges as faces?
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A cube has 6 faces and 12 edges. Triangular prism 5 and 9; square pyramid 5 and 8; triangular pyramid 4 and 6; hexagonal prism 8 and 18.
13 A solid block is built from 3 × 4 × 5 small cubes. How many of the small cubes can be seen from the outside (they have at least one face on the surface)?
Show worked solution
The hidden core is 1 × 2 × 3 = 6 cubes. 60 − 6 = 54 cubes are on the surface.
14 An octagonal pyramid has an octagon for its base.1 It has 9 faces.
2 It has 16 edges.
3 It has 16 vertices.
Which statements are correct?
2 It has 16 edges.
3 It has 16 vertices.
Which statements are correct?
Show worked solution
8 triangles + 1 base = 9 faces (1 ✓). 8 base edges + 8 slanting = 16 edges (2 ✓). 8 base corners + 1 apex = 9 vertices, not 16 (3 ✗). 1 and 2.
15 A 3 × 3 × 3 cube stands on a table. Every face that can be seen is painted — the top and the four sides, but not the bottom. How many small cubes have paint on exactly two faces?
Show worked solution
Top layer: the 4 edge-middles (top + one side). Middle layer: the 4 vertical-edge cubes (two sides). Bottom layer: the 4 corners (two sides, no bottom). 4 + 4 + 4 = 12.