Lesson Guide
Explanation
Count a stack layer by layer from the bottom and add the layers. A solid three by three by three cube has nine in each layer and three layers, twenty-seven; a two by two by two stack is four and four, eight. The one thing never to do is count only the cubes you can see and stop, because the ones hidden at the back are part of the solid too.
Your Turn
How many cubes are in a solid 3 by 3 by 3 cube? Check: 9 × 3 = 27.
Watch Out
Hidden cubes still count in the total. Looking at a picture, at least one cube is usually out of sight.
Explanation
A view is a flat drawing of what you see looking straight at the stack from one direction. The front view shows how tall each column is but says nothing about how many cubes stand behind each one, and the top view shows which squares of the base are covered but nothing about height. Because two different stacks can share a front view, questions usually give you two or three views together. Always draw a view as flat squares on a grid, never as a three-dimensional picture.
Your Turn
Does the front view tell you how many cubes are behind the front row? Check: no.
Watch Out
A view is flat. Drawing it as a 3D sketch answers a different question.
Explanation
The cubes you cannot see are the ones with no face on the surface at all. Peel one layer off every side and what is left is the hidden block. In a three by three by three cube that leaves exactly one small cube buried in the middle, so twenty-six of the twenty-seven have a face showing. In a four by four by four, peeling a layer from each side leaves a two by two by two block, eight hidden cubes.
Your Turn
How many cubes are hidden inside a 4 by 4 by 4 cube? Check: 2 × 2 × 2 = 8.
Watch Out
"Hidden from where you stand" and "no face on the surface" are different. These questions mean the second.
Explanation
When a big cube is painted all over and then split into small cubes, where each small cube sat decides how many of its faces are painted. The eight corner cubes have three painted faces. The cubes along an edge but not at a corner have two, and in a three by three by three there is exactly one such cube on each of the twelve edges, so twelve cubes have exactly two. The six cubes in the middle of a face have one, and the single centre cube has none. Eight plus twelve plus six plus one is twenty-seven, which is the check.
Your Turn
In a painted 3 by 3 by 3 cube, how many small cubes have exactly 3 painted faces? Check: 8, one at each corner.
Watch Out
"Exactly two" means two and no more. The corner cubes have three, so they do not count.
Explanation
One painted cube answers four different questions with four different numbers: twenty-seven altogether, one completely hidden, twelve with exactly two painted faces, twenty-six with at least one. The words decide which count you make, and the three to watch are exactly, at least and altogether. Read them before you start counting. Quiz and exercises now.
Your Turn
In a painted 3 by 3 by 3 cube, how many small cubes have at least one painted face? Check: 27 − 1 = 26.
Watch Out
"At least one" includes the cubes with two and three. Only the buried centre is left out.
📌 Key Points — this lesson in a nutshell
- Count a stack layer by layer, from the bottom up, and add.
- A view is a flat drawing: front, side or top. It shows how many columns are tall, not how many cubes are behind.
- Cubes you cannot see = total cubes − cubes with at least one face showing.
- On a painted cube, a corner cube has 3 painted faces, an edge cube 2, a face middle 1, and the centre none.
Step 1: Introduction
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