# Cubes Here and There

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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This is all about putting green cubes on top of red cubes with some simple rules.

1) The red cubes must be touching the floor (or table top etc.).

2) The green cubes must not be touching the floor.

3) All the cubes are interlocking cubes so they can only be joined square face to square face.

4) The green cubes are next to each other.

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In the pictures above, I used two red cubes and two green cubes, but in different shades of green so as to help me with the method I used to find all the possibilities. You might do it some other way.

You have to look carefully for answers that are really the same - just turned around. So these "other" ones are really the same as the some of the five above.

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Some of these models have green cubes that "hang-over" and for this challenge, we'll decide not to use these. (But you could make the activity harder by including them, if you want!). We'll only use these;

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So, your first challenge is to find the possibilities with two green on three red cubes.

When you've done that, what can you say about the results you might get for having two on four?

Can you give some reasons for your predictions?

How about testing whether they were right?

How do you know you have all the shapes?

Do you have a method for making the shapes?

Are all your shapes different? Are you sure?

So, what will be the number of shapes you can make when you have more red cubes? Why?

Do you have a method for making the shapes?

Are all your shapes different? Are you sure?

So, what will be the number of shapes you can make when you have more red cubes? Why?

Alfie from Grenville Combined School send in this solution.

I think there are 32 solutions in cubes here and there.My method is to write each solution using the letter 'r' as red and 'g' as green and made a picture with each combination.

Perhaps you could send us some of these pictures, Alfie? That would be very helpful.

Here is a very good solution that was sent in by Martin and William (with lots of help from Sebastian) from St Martin's Ampleforth .Well done you three! This is what they wrote and drew:

First of all, we tried to make random shapes according to the rules.

Then we realised that it wasn't working, so we tried making groups of bases. This was more successful so we started making the cubes on top. Once we had finished we had an answer of $39$. However we found that we had duplicated some of the shapes. We got rid of them and our final answer is......$35$, as you can see here.

It was great fun working out the answer to this excellent challenge.

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Is this all that there are or can you find any more?

### Why do this problem?

This problem involves many aspects of mathematics, both content and processes, and it is a good one to introduce investigative approaches. Children will be counting and developing spatial awareness, and you can use it to highlight working systematically, representing and making conjectures.

### Possible approach

In order for children to begin to make conjectures about a mathematical situation, they need to be very familiar with that situation. In this case, learners must be very comfortable with the making of shapes according to the rules. So, this activity would be best introduced as practically as possible with cubes, going through the rules very carefully so that everyone really understands what
is and what is not allowed. Ask pupils to show examples of what is allowed, and when one example is shown that is not allowed, invite children to explain the reasons. You could also use the pictures of the 'two on two' cubes as a stimulus for this.

As learners begin to work on the challenge, perhaps in pairs, it is a good idea to encourage them to record the shapes they make. Emphasise that this recording is for them to remember what they have done and need not (in this particular case) be understood by any adults. They just need to be able to look back and see whether they have made that shape before. You could stop the group after
five minutes or so to share some ways of representing the models that pairs have developed. (If a large number of cubes is available and there are fewer pupils then they can keep each shape that is made and so there is no need to record them.)

Once all the different ways of having two on three have been found, ask children to talk in pairs about their predictions for two on four. Invite them to write up their ideas, with reasons, on a sheet of A4 to come back to later. As the class works on making the different two on four models, you can bring them together at various stages to talk about how they know they will be able to make
them all and whether they want to alter their conjectures based on what they have done so far.

### Key questions

Tell me about the way that you are making these shapes.

How do you know you will have them all?

You look as if you have a method for making more and more shapes - tell me about it.

Is this shape the same or different compared with that one? Why?

So, what will be the number of shapes you can make when you have more red cubes? Why?