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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.


The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Partly Painted Cube

Age 14 to 16
Challenge Level


Why do this problem?

There are many ways of tackling this problem, so there is an opportunity for students to share insights and consider the benefits of the different approaches.

Students have an opportunity to play with algebra they might not normally meet, forming cubic expressions that can be related back to the physical properties of the cuboids they represent.

This problem allows students to model the way mathematicians work, by taking a starting point and then asking their own questions and making conjectures that may lead to generalisations.

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: Partially Painted Cube.

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 members of 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.
You may want to make isometric paper, cubes, poster paper, and coloured pens available for the Resource Manager in each group to collect.

When groups have had a chance to work on the first two questions (this may take more than a lesson!) allow some whole-class time for groups to share their thoughts, questions, and ideas for possible lines of enquiry. There are some suggestions of possible questions to consider in the problem which can be used as prompts for classes who are struggling to come up with conjectures of their own.

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 the different combinations of faces that can be painted - what could you do with this information?
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 effective group work, this list of skills may be helpful (Word, pdf). Ask learners to identify which skills they demonstrated, and which skills they need to develop further.

If your focus is mathematical, these prompts might be useful:

What solid are you left with when you remove the painted cubes? How can you express this algebraically (for 1, 2, 3... faces painted)?


What does this tell you about the factors of the number of unpainted cubes?
There is only one way to end up with 45 unpainted cubes. Are there any numbers of cubes you could end up with in more than one way?
How can you convince yourself that it is impossible to end up with 50 unpainted cubes?

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.

Painted Cube provides a suitable introduction to some of the ideas met in this problem.

Possible extension

Devise a method for determining whether any given number of unpainted cubes can arise from painting whole faces of a large cube, and if so, how it can be done.