Counting cubes
A drawing of stacked cubes asks how many there are, including the ones you can't see. Here's how to count every one, every time.
Rush doesn’t practise counting cubes yet. These games practise related skills: Top View and Shapes & Space.
Chapter 1
What are counting cubes questions?
You see a drawing of cubes stacked on a flat base. It’s drawn from the front, a little to the right and from above, so you see the fronts, the right-hand sides and the tops.
Most questions ask how many cubes make the model. That means every cube, including the ones hidden behind and underneath the others. Some ask how many can’t be seen, or how many more would make a solid block. They all start with the same careful count.
How many cubes make up this model?
- 10
- 11
- 12
- 14
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Every cube is in a stack
A stack is a pile of cubes standing on one square of the floor. Count each stack from its top cube down to the floor, then add: 1 + 3 + 2 = 6.
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Cubes can’t float
A cube up in the air has cubes under it, all the way to the floor. Here you can see 4 cubes, but the top one stands on a fifth, hidden at the back. So there are 5.
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One cube, up to three faces
A cube can show its top, its front and its right side. That’s three faces but only one cube, so count stacks, never faces.
Chapter 2
Solve it in 4 steps.
Here’s the question. Draw the grid on your paper as you go: it keeps the count straight.
The question
How many cubes make up this model?
- 10
- 11
- 12
- 14
-
Step 1
Draw the floor plan.
Each stack stands on one square of the floor. Draw a 3 by 3 grid with the back row at the top. Where you can see the floor, there’s no stack: put a dash there.
The floor shows at the front middle: a dash.
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Step 2
Fill in the stacks you can see to the floor.
If a stack’s front or right side shows all the way down, count down it. That’s the front row (1 and 1), the middle stack (1, its front shows over the gap), the middle right (1) and the back right (2).
Five stacks you can count from top to floor.
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Step 3
Bottom hidden? Add the stack in front.
Count the cubes that show above the stack in front, then add that stack’s height. Middle left: 1 shows above the front cube, so 1 + 1 = 2. Back left: 1 shows above that, so 1 + 2 = 3. Back middle: only its top shows, level with the middle, so 0 + 1 = 1.
The 2 hidden cubes, drawn see-through.
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Step 4
Add up, then check by layers.
The rows make 6 + 4 + 2 = 12. Now count again from the bottom: every stack has a cube on the floor (8), 3 stacks reach the second layer and 1 reaches the third. 8 + 3 + 1 = 12, so it’s right.
1on layer 3
3on layer 2
8on the floor
8 + 3 + 1 = 12, the same as the grid.
Three mistakes to avoid
| The mistake | Do this instead |
|---|---|
| Counting only the cubes you see. 10 cubes show. Two more are hidden at the bottom of the back-left and middle-left stacks. | Count every stack from its top right down to the floor. |
| Reading two tops as one. 11 misses a stack. The back-middle top sits just behind the middle top, at the same height, so they look like one long top. | Give every top its own square on the grid. |
| Adding the hidden cubes again. 14 counts the 2 hidden cubes twice. Counting each stack to the floor already included them. | Count each stack once, top to floor. |
So it’s 12. The grid adds up to 12, and the layers give 8 + 3 + 1 = 12 too.
Remember: Every top, down to the floor.
Chapter 3
When it gets harder.
Taller models, and the other questions papers ask about them. The grid answers them all.
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A tower behind a tower
Work from the front to the back.
The middle stack shows 2 cubes above the front cube: 2 + 1 = 3. The tower shows 1 above that: 1 + 3 = 4. Trap: the tower looks as if it sits on the middle stack, so it gets counted as 1.
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A solid block: multiply
Wide × deep × tall.
This block is 4 × 3 × 2 = 24 cubes, but only 18 show. Trap: counting the cubes you can see. Even a block 2 cubes each way hides one of its 8.
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How many can’t be seen?
Count the model, then take away the cubes that show.
This model has 6 + 4 = 10 cubes, and 7 show at least one face. 10 − 7 = 3 can’t be seen, all under the back stacks. Trap: answering 7, the ones you can see.
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How many more make a block?
The block’s total, take away the model.
A block 3 wide, 2 deep and 2 tall has 12 cubes. The model has 8, so it needs 4 more. Check: three places are short, by 1, 1 and 2. Trap: 3, one cube for each place.
Chapter 4: In the exam
Bigger models, five close answers.
In practice papers the models have 13 to 19 cubes, up to 4 of them hidden, and the five answers are only 1 to 3 apart. One missed cube picks a wrong answer, so always check.
How many cubes make up this model?
- A14
- B15
- C16
- D17 (the answer)
- E18
- Front: 1, 1, a dash, 1.
- Middle: the left two show only their tops, level with the front: 1 and 1. Then 1, and 2 down the right side.
- Back: what shows above the stack in front, plus that stack: 1 + 1, 2 + 1, 1 + 1, then 2 down the side.
- Add: 9 + 5 + 3 = 17. Layers: 11 + 5 + 1 = 17.
So it’s D, 17. Only 14 cubes show, so option A is the trap. The other 3 hide at the bottom of the back row.
Remember: Count it twice: stacks, then layers.
Also in the exam
- “How many cubes touch the table?” That’s the bottom layer: one cube for every stack, so every square on your grid without a dash. That’s 11 in the model above.
- “How many cubes were taken away from a solid block?” The same as how many more: the block’s total take away the model.
- A rule that cubes can’t hover. Some papers print it. It’s step 3: every cube you see stands on a full stack, so count the hidden cubes that hold it up, and no others.
Chapter 5
Try it yourself.
Easier ones first. Try each one before you open its answer.
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Question 1Level 1
How many cubes make up this model?
- 6
- 7
- 8
- 9
Show answer
Answer: 8.
How to answer it
- Back row. The back-left stack is the tall one: two cubes show above the front-left cube, and the front-left cube hides one more. 2 + 1 = 3.
- The back-middle stack stands behind the empty place, so all of it shows: 2. The back-right cube's top is level with the cube in front of it: 1.
- Front row: 1, an empty place, 1.
- 3 + 2 + 1 + 1 + 1 = 8 cubes.
Why the other answers are wrong
- 6That counts the squares facing you, as if the model were one cube deep. The bottom cube of the back-left stack and the back-right cube sit behind other cubes.
- 7These are the cubes you can see. The tall stack looks as if it stands on the front-left cube, but it is a row further back, and its own bottom cube is hidden.
- 9That adds the hidden cube again after counting the back-left stack as 3, which already includes it.
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Question 2Level 1
How many of the cubes in this model can't be seen?
- 0
- 1
- 5
- 6
Show answer
Answer: 1.
How to answer it
- Count the model first. Back row 2 and 1, middle row 1 and 1, front row 1 and an empty place: 6 cubes.
- Now look for cubes with no face showing. The back-left stack is 2 tall, but only its top cube shows.
- Its bottom cube is behind the middle-left cube and beside the back-right cube, so no face of it shows. Every other cube shows its top: 1 cube can't be seen.
Why the other answers are wrong
- 0The back-left stack is 2 tall, so something holds up its top cube. That bottom cube has no face showing.
- 5That is how many cubes you can see. The question asks for the ones you can't.
- 6That is every cube in the model.
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Question 3Level 1
How many cubes make up this model?
- 4
- 6
- 7
- 8
Show answer
Answer: 7.
How to answer it
- Back row. The back-left and back-right cubes stand alone, each with its front showing: 1 and 1.
- The back-middle stack is the tall one. One cube shows above the front-middle stack, which is 2 tall: 1 + 2 = 3.
- Front row: only the middle stack, 2 tall.
- 1 + 3 + 1 + 2 = 7 cubes.
Why the other answers are wrong
- 4That counts one cube per stack. The back-middle stack is 3 tall and the front-middle one 2.
- 6These are the cubes you can see. The tall back stack looks as if it sits on the front stack, but it stands a row behind, and its bottom cube is hidden.
- 8That counts the hidden cube twice. Counting each stack down to the floor already includes it.
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Question 4Level 2
How many cubes make up this model?
- 10
- 12
- 13
- 18
Show answer
Answer: 13.
How to answer it
- Back row. The back-right stack shows two cubes down its right side, and the tops of the other two back stacks are level with it: 2, 2, 2.
- Middle row. The left top is level with the back row: 2. The middle cube's front shows over the empty place: 1. The right stack shows two cubes down its side: 2.
- Front row: 1, an empty place, 1.
- 6 + 5 + 2 = 13 cubes.
Why the other answers are wrong
- 10These are the cubes you can see. Three are hidden: the bottom cubes of the back-left, back-middle and middle-left stacks.
- 12This reads the back-left stack as one cube. Only its top shows, but that top is level with the 2-high stacks beside it and in front of it, so it is 2 tall.
- 18A solid block 2 cubes tall would have 18. The front-middle place is empty and three stacks are only 1 tall.
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Question 5Level 2
How many of the cubes in this model can't be seen?
- 3
- 4
- 9
- 13
Show answer
Answer: 4.
How to answer it
- Go stack by stack, looking for cubes with no face showing.
- Back-left: 3 tall, and only its top cube shows above the middle-left stack. 2 hidden.
- Back-middle: 2 tall, and only its top cube shows above the middle cube. 1 hidden. Back-right: both cubes show down its right side.
- Middle-left: 2 tall, and its bottom cube is behind the front-left cube. 1 hidden. Every other stack is 1 tall and shows its top.
- 2 + 1 + 1 = 4 cubes can't be seen.
Why the other answers are wrong
- 3This finds the hidden cubes in the back row but misses the one under the middle-left stack, behind the front-left cube.
- 9That is how many cubes you can see.
- 13That is every cube in the model.
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Question 6Level 2
How many more cubes are needed to turn this model into a solid block 3 cubes wide, 3 deep and 3 tall?
The solid block: 3 × 3 × 3
- 8
- 12
- 15
- 18
Show answer
Answer: 15.
How to answer it
- A solid 3 by 3 by 3 block has 27 cubes.
- Back row. The left cube's top is level with the cube in front of it: 1. The middle stack is the tall one: one cube shows above the middle stack in front, which is 2 tall, so 1 + 2 = 3. The right stack shows two cubes down its side: 2.
- Middle row: 1, then 2 (one cube shows above the front-middle cube), then 1.
- Front row: 1, 1 and an empty place. 6 + 4 + 2 = 12.
- 27 − 12 = 15 more cubes.
Why the other answers are wrong
- 8That is one cube for each place that is short of 3. Most places need 2 cubes, and the empty one needs 3.
- 12That is how many cubes the model has now.
- 18That is 27 take away only the 9 cubes you can see. Three more are hidden inside the model, so they are already there.
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Question 7Level 3
How many cubes make up this model?
- 14
- 17
- 18
- 22
Show answer
Answer: 18.
How to answer it
- Back row. The back-left tower shows one cube above the middle-left stack. The middle-left stack shows two cubes above the front-left cube, so it is 2 + 1 = 3, and the tower is 1 + 3 = 4.
- The back-middle stack shows one cube above the middle cube: 1 + 1 = 2. The back-right stack shows three cubes down its right side: 3.
- Middle row: 3 on the left (worked out above). The middle cube shows only its top, level with the front-middle cube: 1. The right stack shows two cubes down its side: 2.
- Front row: 1, 1, 1.
- 9 + 6 + 3 = 18 cubes.
Why the other answers are wrong
- 14These are the cubes you can see. Four are hidden: two under the back-left tower, one under the back-middle stack and one under the middle-left stack.
- 17This misses the middle cube. Only its top shows, just behind the top of the front-middle cube and at the same height, so it can pass for part of that cube.
- 22That adds the 4 hidden cubes to a count that already had them.
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Question 8Level 3
How many cubes make up this model?
- 11
- 16
- 19
- 27
Show answer
Answer: 19.
How to answer it
- A staircase. Back row: the back-right stack shows two cubes down its right side: 2.
- The back-middle stack rises one cube above the back-right stack: 3. The back-left stack rises one above that: 4.
- Middle row, the same way from the right: 1, then 2, then 3. Front row: 2 on the left, then 1 and 1.
- 9 + 6 + 4 = 19. Check by layers: 9 stacks on the floor, 6 reach the second layer, 3 the third and 1 the fourth. 9 + 6 + 3 + 1 = 19 cubes.
Why the other answers are wrong
- 11These are the cubes you can see. Eight are hidden inside the staircase, three of them under the back-left tower.
- 16This reads the back-left stack as a single cube, because only its top cube shows. It rises one above the back-middle stack of 3, so it is 4 tall.
- 27That adds the 8 hidden cubes to a count that already had them.
For parents.
What parents ask us about counting cubes.
Questions parents ask
Do you count the cubes you can't see?
Yes, unless the question asks only for the ones you can see. A cube can't float, so every cube you can see stands on a full stack that reaches the floor. Count those hidden cubes once, as part of their stack.
What if the drawing hides part of the model completely?
The question should say what to assume, for example that there are no gaps, or that hidden cubes are only there to hold up the ones you can see. If it says nothing, count only the cubes needed to hold up every cube you can see. The drawings on this page show every top and every empty place, so each has one answer.
What helps a child who keeps getting the count wrong?
Build models. Stack a few toy bricks or sugar cubes, look at them from the front corner, and ask your child to count before touching anything. Then take the model apart a layer at a time to check.
Is counting cubes the same as finding volume?
For a solid cuboid, yes: length × width × height gives the number of cubes, so a block 4 by 3 by 2 has 24. A model with steps and gaps has no formula, which is why it is counted stack by stack. The formula comes back when a question asks how many more cubes would make a solid block.
Practise the related skills in Rush.
- Top ViewIts block models are drawn from the same angle as these, and choosing the right plan means working out which row each stack stands in, past the stacks in front of it.
- Shapes & SpaceIts volume questions multiply length, width and height, which is the first step in every question that asks how many more cubes would make a solid block.
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