## Six Ten Total

Take sixteen dice - six of one colour and ten of another.

Here we use blue and red.

**Warm up:** check the total of the faces showing is 84 when the dice are arranged as shown:

**To play the Six Ten Total challenge:**

- all the blue need to have the same number on the top
- all the red need to have the same number on the top
- there always needs to be six of one colour and ten of the other
- the red dice have to use a different number to the blue
- there is always a difference of two between the numbers on the blue and red dice.

**THE MAIN CHALLENGE **

What are the possible arrangements when you choose your own numbers for the dice using the rules above?

What is the total for each of these arrangements?

What do you notice about your arrangements and the corresponding totals?

Explain what you notice. What else do you notice?

Can you prove any of the things you've noticed from the main challenge are always true?

Instead of having a difference of two between the numbers showing on each face of the blue and the red dice, choose a new difference.
Can you predict - without having to make them - what numbers need to be on the faces that you can see on the blue and the red dice to make a total of 42?

Are there more ways to make a total of 42?

Explain your reasoning.

What happens if you change the number of blue and red dice yet keep the structure the same? It must be a square and both sets of dice must still make triangles.

Can you make 42?

Explain your reasoning.

*This problem featured in a preliminary round of the Young Mathematicians' Award 2014.*
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### Why do this problem?

This problem gives pupils the opportunity to use their skills of addition and multiplication in a problem-solving situation. It is also useful in expecting learners to persevere while exploring. It also requires pupils to explain what they notice and, when taking it further, to produce a proof.

Possible approach

It would be good to have 16 dice for each pair/group to use to duplicate the given situation. Then ask them about getting the total and how they achieved that. (This can be very enlightening as to how the children, faced with an unusual calculation go about getting a solution.) The rules need to be very clearly stated and checked with the pupils and then they can investigate other
arrangements.

Some may feel that they can quickly move on to recording their arrangements and so may not need the dice.

### Key questions

There can be many opportunities for the questions, "What have you noticed?", "Tell me about this" , "What will you do next?".

### Extension

The final paragraph can be used for this/these pupil/s.

### Possible support

Some pupils may need help in keeping the dice together. A frequent reminder by asking "What were the rules?" may be necessary.