The box that two sizes of cube fill exactly
Where to find it
This is a worded PSLE question, so we don’t reproduce it here. Find it in your Ten-Year Series (TYS) or the official paper — 2019 Paper 2, Q15, parts (a) and (b) (the volume question about packing two sizes of cube) — then follow our worked solution below.
The lock
A child counts the first layer and never notices the hinge: the two cube sizes share the box's height exactly, which only works if 2 large edges equal 3 small edges. Miss that and part (a) is a guess.
The key
Read the geometry off the first layer, then use the volume split. The figure is doing the teaching.
Worked steps
- The first layer shows 2 large cubes spanning the same height as 3 small cubes, so 2 large edges \(=\) 3 small edges.
- The 8 large cubes make \(8 \div 2 = 4\) large layers; that same height fits 6 small layers, each of 6 small cubes, so small cubes \(= 6 \times 6 = 36\).
- (b) Large cubes' volume \(= \frac{3}{7} \times 4032 = 1728\) cm³; small cubes' volume \(= 4032 - 1728 = 2304\) cm³.
- One small cube \(= 2304 \div 36 = 64\) cm³, so its edge \(= \sqrt[3]{64} = 4\) cm.
Answer: (a) 36 small cubes. (b) 4 cm.
What makes it click. Once "two large equals three small" is on the page, the count of small cubes is forced, and the cube root at the end is the gentle part. Check: each large cube is \(1728 \div 8 = 216\) cm³, edge 6 cm, and \(2 \times 6 = 3 \times 4\).
Independently solved, matches the GPA marking-scheme key. Open the full worked solution →
