The idea in brief. A cube has 6 faces. Take one away for every cube glued to it, and one more if it is standing on the table and the table faces are not painted. There is no position rule: these solids are irregular and where a cube sits in one decides nothing.
Lesson video, with the working
Painted cubes
3 min 33 s
The film teaches 2 questions in turn. Each question shows first; its working is written as the film explains it.
Seven unit cubes are glued together to form the solid shown. The outside of the solid is painted, but the faces resting on the table are not painted. How many of the seven cubes have exactly 4 painted faces?
Cubes with 4 painted faces = 4 + 1 = 5
Ans: 5
Example 1
Ten unit cubes are glued together to form the staircase shown. It is one cube deep, and its four columns are 4, 3, 2 and 1 cubes high. The outside is painted, but the faces resting on the table are not painted. How many of the ten cubes have exactly 4 painted faces?
Cubes with 4 painted faces = 1 + 1 + 1 = 3
Ans: 3
How to solve it
Start every cube at 6 faces.
Count the joins. A face glued to another cube is not painted, and one join hides 2 faces.
Count the faces on the table. A face resting on the table is not painted either.
Subtract for each cube: an arm cube has 6 minus 1 join minus 1 for the table, which is 4 painted faces.
Check every cube in turn: the lower stacked cube has 2 joins, so 6 minus 2 is 4 painted faces.
Every cube starts at 6 faces. A face glued to another cube is not painted. A face resting on the table is not painted.
Practice questions with worked solutions
Question 1 is free to try. The other 5 open in the free trial, each with its full working.
the left place has cubes 1 high, the middle place 3 high and the right place 1 high
front view squares = 1 + 3 + 1 = 5
the front view is a column of 3 with 1 square on each side of its bottom square
Answer: 5 unit squares: a column of 3 with 1 square on each side of its bottom square
For parents
How do you count faces in painted cube questions in P6?
Unit cubes are glued into a solid, the outside is painted, and the question asks how many faces are painted, or how many cubes have a given number of painted faces.
What your child learns here
Unit cubes are glued into a solid, the outside is painted, and the question asks how many faces are painted, or how many cubes have a given number of painted faces. This builds on building solids with unit cubes from earlier years and applies it to a Primary 6 counting question.
Start every unit cube at 6 faces.
Take off 2 faces for every place where two cubes are glued together.
Take off 1 face for every cube resting on the table, when the base is not painted.
Check cube by cube when the question asks how many cubes have exactly a certain number of painted faces.
Turn a count of painted faces into an area by multiplying by the area of one face.
Does the answer make sense?
The painted faces can never be more than 6 times the number of cubes, since every join and every face on an unpainted table takes some away. A count of glued faces your child gives must be even, because each join hides one face on each of two cubes.
How to practise these questions
The Basic questions train counting painted faces on a small solid. Intermediate questions add leaving the faces on the table unpainted and counting the cubes that show a stated number of painted faces. Challenge questions count the faces paint cannot reach, remove a cube and paint again, or find a painted area when each cube has an edge measured in centimetres. If counting painted faces gets confusing, read the written step-by-step solution together.
Questions parents ask
How do you count the painted faces of joined cubes? Start with 6 faces for every cube. Take off 2 for each place where two cubes are glued, and 1 for each cube standing on the table if the base is not painted.
Why does one join hide two faces? Where two cubes are glued, each cube has a face pressed against the other. The brush cannot reach either face, so one join hides one face on each cube.
Which syllabus topic do painted cube questions belong to? They apply building solids with unit cubes, which the MOE syllabus places under Volume of Cube and Cuboid in Primary 5. This page uses that skill in a Primary 6 counting question.
