Concept 7 of 9
Cube Counting: Layer by Layer
Today's big idea
Count a stack one flat layer at a time, including the cubes buried out of sight.
Stack split into bottom, middle and top so each layer is counted once.
How it works
👀 What's going on?
A photo of a stack of cubes only ever shows you the outside. Some cubes are buried behind and beneath the ones you can see, holding them up, and those count too.
💡 Here's the trick
Break the stack into flat layers and count one layer at a time, starting from the bottom. A cube resting on nothing is impossible, so any cube with something above it must be there even when you cannot see it.
🪄 How to do it
- 1.Count the cubes in the bottom layer of the stack in the picture above.
- 2.Count each layer above it in turn.
- 3.Add all the layers together.
✍️ Try it
One stack has 9 cubes on the bottom, 4 in the middle and 1 on top. Another is a solid block, 3 cubes wide, 3 cubes deep and 3 cubes tall. How many cubes in each?
The first is 9 plus 4 plus 1, which comes to 14. The second has three layers of 9, which comes to 27.
⚠️ Watch out
Count what must be underneath, not only what you can see. A cube sitting high in the stack needs a whole column of cubes below it, and those buried ones are what most children forget.
🌟 Why this matters
Stacking is everywhere. Boxes in a delivery van, crates in a shop store room, and bricks part-way up a wall all hide their inside layers from you.
Exam tip: Write each layer's count down the side of the figure, because holding three numbers in your head is where one of them goes missing.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A stack has bottom 9 cubes, middle 4, top 1. How many cubes?”
- 1
Count each layer
Bottom = 9. Middle = 4. Top = 1.
- 2
Add all layers
9 + 4 + 1 = 14
- 3
Answer ✓
= 14 cubes
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap each layer to count its cubes. Add them all up for the total.
STACK · ROUND 1 OF 3
Quick check
✅ VERIFICATION
A solid block is 3 cubes wide, 3 deep and 3 tall. How many small cubes cannot be seen from ANY side?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Dice
Bottom = 7 − top. 1↔6, 2↔5, 3↔4.