Painted cube
Problem
Painted Cube printable worksheet
Imagine a large cube made up from $27$ small red cubes.
Imagine dipping the large cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into its small cubes.
How many of the small cubes will have yellow paint on their faces?
Will they all look the same?
Now imagine doing the same with other cubes made up from small red cubes.
What can you say about the number of small cubes with yellow paint on?
Click here for a poster of this problem.
Getting Started
How many vertices, faces and edges are there in a cube?
Where are the cubes with no faces painted?
Where are the cubes with 1, 2, 3, 4, 5, 6 faces painted?
Student Solutions
We received many correct solutions to this problem - from Adam from Knoxdale P. S., Ramsudheer from Smith Elementary School, Ojaswi, Bill from King's St. Alban's, Chantell, Billy, Connor and Chanel from Arunside, Rajeev from Fair Field Junior School, Jessica from Beeston Rylands Junior School, Finlay from Gledhow Primary School, Olivia and Martha from St.Johns C.E. Primary School, Yuki from King's School Ely, Belky, Mark from Gledhow Primary School, James and Arjun from Wilson's, William from Barnton Community Primary School Alice, Joanna from Woodmill High School, Derek, Robert from Ardingly College Junior School, Joe from Lady Manners School in Bakewell and students from Rawdon Littlemoor Primary School. Well done to you all.
Alan wrote:
Chantell, Billy, Connor and Chanel sent us this clear table summarising their findings. Rajeev sent us a very similar table of results.
Yuki noticed that:
Students from Crestwood College also mentioned this.
Arjun also worked on this:
Mark from Gledhow Primary School produced this spreadsheet.
Robert summarised his findings as follows:
"Let n= number to be cubed
No. of small cubes with 6 red faces = (n-2)$^3$
This is because the cubes in the centre remain clean, so you must take one off either edge and then cube it.
No. of small cubes with 5 red faces = 6(n-2)$^2$
This is because there are 6 faces to the cube, and only the ones not on the edge remain clean on 5 sides. So you must take one off either edge, then square it, then multiply it by the 6 faces.
No. of small cubes with 4 red faces = 12(n-2)
This is because there are 12 non-corner edges which is multiplied by n (which = 1 whole edge) -2 for the 2 corners.
No. of small cubes with 3 red faces = 8
This is because there are always 8 corners
The total No. of small cubes is always n cubed"
Alice described her findings in a similar way:
"First of all I imagined a 3x3x3 cube being dipped in paint - that's how I worked the first one out then we worked out that
1) the number of cubes with 6 red faces equalled (n-2)$^3$, like when you take the skin off a square orange, taking a layer off each side
2) 5 red faces 6(n-2)$^2$, which is like the above but for the faces instead of the middle (it's squared not cubed ) and you have to multiply it by 6 because there are 6 faces
3) 4 red faces 12(n-2) this is the edges, take 2 for the corners and multiply it by 12 because that's how many edges there are
4) 3 red faces always got to be eight because these are corners (unless your cube is 1x1x1)
5) total number of small cubes is n$^3$
Joe, Derek and Alice completed the table of results:
|
Size of large cube |
No. of small cubes with 6 red faces |
No. of small cubes with 5 red faces |
No. of small cubes with 4 red faces |
No. of small cubes with 3 red faces |
Total No. of small cubes |
|
3 x 3 x 3 |
1 |
6 |
12 |
8 |
27 |
|
4 x 4 x 4 |
8 |
24 |
24 |
8 |
64 |
|
5 x 5 x 5 |
27 |
54 |
36 |
8 |
125 |
|
6 x 6 x 6 |
64 |
96 |
48 |
8 |
216 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
10 x 10 x 10 |
512 |
384 |
96 |
8 |
1000 |
|
23 x 23 x 23 |
9261 |
2646 |
252 |
8 |
12167 |
|
n by n by n |
(n-2)$^3$ |
6(n-2)$^2$ |
12(n-2) |
8 |
n$^3$ |
NB. The values for n are correct unless n = 1.
When n = 1 the single cube will have no red faces.
In her conclusion Alice added that in the second column the numbers were all cubic numbers, in the third column square numbers times 6 and in the fourth column multiples of 12.
Belky explained how she reached her results. This is how she arrived at a formula for the number of cubes with two faces painted :
Both William and Derek showed that:
We can verify that all cubes have been accounted as:$$(n-2)^3 + 6(n-2)^2 + 12(n-2) + 8 $$ $$= (n-2) (n^2 - 4n + 4) + 6 (n^2 - 4n + 4) + 12n - 24 + 8 $$ $$ = n^3 - 4n^2 + 4n - 2n^2 + 8n -8 + 6n^2 - 24n + 24 + 12n - 24 + 8 $$ $$ = n^3 $$
Derek added this diagram to support his conclusions:
Mollie and Hannah from Comberton Village College also sent in some excellent solutions; follow the links below to see their work:
Mollie's solution
Hannah's solution
Teachers' Resources
Why do this problem?
Exploring a variety of painted cubes may produce patterns which students can describe spatially, numerically and algebraically. Students can appreciate the benefits of keeping a clear record of results, and apply their insights from the first case to ask themselves questions about further cases.
This problem lends itself to collaborative working, both for students who are inexperienced at working in a group and students who are used to working in this way.
Many NRICH tasks have been designed with group work in mind. Here we have gathered together a collection of short articles that outline the merits of collaborative work, together with examples of teachers' classroom practice.
Possible approach
This printable worksheet may be useful: Painted Cube.
This approach could be used if you want the focus to be on the mathematical structure of the problem:
"Here is a 3 by 3 by 3 cube made up of 27 smaller cubes. Imagine I dipped it in a pot of yellow paint so that each face of the large cube was covered. Then after the paint has dried, imagine I split it into the 27 original small cubes. Can you work out how many cubes will have no paint on them? How many will have just one face painted? Or two faces painted, and so on."
Give students time to discuss with their partners and work out their answers. While they are working, circulate and observe the different approaches that students are using, and challenge them to explain any dubious reasoning.
Bring the group together to share their responses. Collect the answers for the number of cubes with 0, 1, 2, 3... faces painted, and note that they add up to 27. Invite students to explain how they worked it out.
"I'd like you to work on some cubes of different sizes until you are confident that you can always work out how many cubes will have 0, 1, 2 and 3 faces painted. In a while, I'll be choosing a much larger cube at random, and you?ll need to have an efficient method of working it out."
Here are some prompts that could be used if students get stuck:
"Where are the cubes with no faces painted?"
"Where are the cubes with 1, 2, 3 faces painted?"
"How many of each type of cube would you have in a 4 by 4 by 4? in a 7 by 7 by 7? in an n by n by n cube?"
Bring the class together and challenge them to explain how they can work out the number of cubes of each type in a 10 by 10 by 10 painted cube. Depending on the students' experience of working with algebra, you could work together on creating formulas for the number of cubes of each type in an n by n by n cube. The Solution contains a table that shows the results very clearly.
This approach could be used if you want the focus to be on developing groupwork:
This is an ideal problem for students to tackle in groups of four. Allocating these clear roles (Word, pdf) can help the group to work in a purposeful way - success on this task should be measured by how effectively the group work together as well as by the solutions they reach.
Introduce the four group roles to the class. It may be appropriate, if this is the first time the class have worked in this way, to allocate particular roles to particular students. If the class work in roles over a series of lessons, it is desirable to make sure everyone experiences each role over time. For suggestions of team-building maths tasks for use with classes unfamiliar with group work, take a look at this article and the accompanying resources.
Hand out this task sheet (Word, pdf) to each group, and make it clear that everyone needs to be ready to share their findings with the rest of the class at the end of the sessions. Exploring the full potential of this task is likely to take more than one
lesson, with time in each lesson for students to feed back ideas and share their thoughts and questions.
You may want to make isometric paper, cubes, poster paper, and coloured pens available for the Resource Manager in each group to collect.
While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together.
You may choose to focus on the way the students are co-operating:
Group A - Good to see you sharing different ways of thinking about the problem.
Group B - I like the way you are keeping a record of people's ideas and results.
Group C - Resource manager - is there anything your team needs?
Alternatively, your focus for feedback might be mathematical:
Group A - I like the way you are considering the structure of the cube.
Group B - You've identified which cubes end up with one face painted - can you think of a way of quickly counting them?
Group C - Good to see that someone's checking that the answers are in line with your predictions.
Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.
We assume that each group will record their diagrams, reasoning and generalisations on a large flipchart sheet in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:
- Every group is given a couple of minutes to report back to the whole class. Students can seek clarification and ask questions. After each presentation, students are invited to offer positive feedback. Finally, students can suggest how the group could have improved their work on the task.
- Everyone's posters are put on display at the front of the room, but only a couple of groups are selected to report back to the whole class. Feedback and suggestions can be given in the same way as above. Additionally, students from the groups which don't present can be invited to share at the end anything they did differently.
- Two people from each group move to join an adjacent group. The two "hosts" explain their findings to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.
Key questions
If your focus is mathematical, these prompts might be useful:
Where are the cubes with no faces painted?
Where are the cubes with 1, 2, 3, 4, 5, 6 faces painted?
How many of each cube will you have in an n by n by n cube?
How do your algebraic expressions relate to the geometry of the situation?
Possible support
By working in groups with clearly assigned roles we are encouraging students to take responsibility for ensuring that everyone understands before the group moves on.
Possible extension
Partly Painted Cube provides a suitable follow-up activity.