Maths olympiad question · P6★★★★ Expert
Solids: olympiad question 12
An expert question on solids for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 first rounds and finals: painted small cubes, water levels and the surface area of stacks.
The question
125 small cubes with 1 cm edges make a big cube, 5 along each edge.
Through the middle of the faces, front to back and left to right, a straight square tunnel 1 cm by 1 cm is cut right through (a whole row of small cubes taken out); the 2 tunnels cross at the centre (shown).
With the tunnels cut, what is the surface area of what is left (tunnel walls included), in cm²?
Show the hints (3)
- Hint 1: Split it in two: what do the six outer faces lose, and what do the tunnels add? Count in small squares (one face of a small cube).
- Hint 2: Each tunnel opens two holes outside. Inside, count walls cell by cell: a tunnel cell away from the centre has a wall on each of its 4 sides; the centre cell is shared by the tunnels and has fewer.
- Hint 3: Outside: 6 × 5 × 5 − 4 = 146 squares. There are 2 × 4 = 8 tunnel cells away from the centre.
Worked solution
Have a go first, then read the solution step by step.
- 1.Count in small squares (1 × 1 = 1 cm² each). The outside had 6 × 5 × 5 = 150; the 2 tunnels open 2 holes each, leaving 150 − 4 = 146.
- 2.Inside, cell by cell: each tunnel has 5 cells; apart from the centre there are 2 × 4 = 8, each with a wall on all 4 sides: 8 × 4 = 32 squares.
- 3.The centre cell is shared by the 2 tunnels: of its 6 faces, 4 open into tunnels, leaving 6 − 4 = 2 walls. Inside the tunnels: 32 + 2 = 34 squares.
- 4.The whole surface: 146 + 34 = 180 squares.
- 5.Surface area = 180 × 1 = 180 cm².
- ⚠️Common mistake: giving every tunnel a full 4 × 5 wall squares, so the centre's walls are counted twice or counted when they aren't there; or not taking off the holes outside. And multiply by each square's 1 cm².
Answer: 180 cm²
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