Counting Cubes Puzzles
How many cubes are in the stack? The ones you can't see still count.
Practice
How many cubes are in this solid? Every cube rests on the floor or on another cube.
Pick A, B, C or D.
Count stack by stack
The usual mistake is counting the cubes you can see. In a stack, some cubes are hidden under or behind others. The fix is to count columns, not cubes:
- Look at the floor and find each square that has cubes on it.
- For each one, find the top cube and count down to the floor: that is the stack's height.
- Add up the heights.
Worked example
A medium stack. Count it stack by stack, choose an answer, then open the answer to check your total.
How many cubes are in this solid? Every cube rests on the floor or on another cube.
Show the answer and why
The answer is B. Add the stacks: 2 + 1 + 2 + 1 + 1 + 2 + 2 + 2 = 13. Cubes under the tops still count.
- A: Count again stack by stack: the total is 13.
- C: Count again stack by stack: the total is 13.
- D: Count again stack by stack: the total is 13.
A faster check: write the heights on a map
For bigger stacks, sketch the floor as a small grid, the way it looks from above, and write each stack's height in its square. Then add up the grid. This is the same picture as a top view with numbers on it, and it stops you counting the same stack twice or skipping one at the back.
Common mistakes
- Counting only what you can see. Counting visible cubes misses the ones inside the stack. Count every cube in each column, including those under its top cube.
- Counting faces instead of cubes. One cube can show three faces at once: top, front and side. Count each cube once, by its stack.
- Losing your place. Work in a fixed order, back row first, left to right, so every stack is counted exactly once.
- Misreading height. Count the cubes in the stack from its top cube down to the floor, not the gap between stacks.
How the levels get harder
| Level | Floor | Max stack height | Cubes |
|---|---|---|---|
| Easy | 3×3 | 2 | 5 to 10 |
| Medium | 3×3 | 3 | 8 to 16 |
| Hard | 4×4 | 3 | 12 to 26 |
| Expert | 4×4 | 4 | 16 to 34 |
From Medium up, at least two cubes in every puzzle are hidden from view, so counting what you can see is never enough.
Where counting cubes questions come up
Block counting is a classic spatial ability item, and it shows up in aptitude tests, school entrance tests and some job assessments. It is also how volume is first taught in school: the US Common Core standard for grade 5 (5.MD.C.4) asks students to measure volume by counting unit cubes. Practising on stacks where some cubes are hidden trains the step that matters most, picturing what is behind and underneath.
Why every answer is certain
In every puzzle the top of each stack is visible. Since cubes can't float, a visible top cube tells you exactly how tall its stack is, and the total can only be one number.
Counting cubes questions
Do hidden cubes count?
Yes. Every cube sits on the floor or on another cube, so a cube you can see at the top of a stack means every cube below it is there too, even if you cannot see them.
What is the quickest way to count?
Count stack by stack. For each square of the floor, find the top cube and read how tall the stack is, then add the heights. Counting faces or visible cubes misses the hidden ones.
Can a cube be completely hidden?
In these puzzles, every stack's top is visible, so every cube can be worked out. That is what makes each answer certain.
Are counting cubes questions used in tests?
Yes. Block counting appears in spatial ability tests, school maths (volume with unit cubes) and some job aptitude tests.
More 3D practice: cube nets, mental rotation and the 3D puzzle game.