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# Six Ten Total

## Six Ten Total

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

**The main challenge**

#### Taking it further

#### Final challenge

### Why do this problem?

### Possible approach

### Key questions

### Possible extension

### Possible support

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Age 7 to 11

Challenge Level

- Problem
- Student Solutions
- Teachers' Resources

Take sixteen dice - six of one colour and ten of another. Here we have chosen six blue dice and ten red dice.

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

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

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?

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.

Now, what totals can you find? What do you notice? Explain what you notice.

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.*

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.

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 finding 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.

What have you noticed?

Tell me about this.

What will you do next?

The final paragraph can form an extension for pupils who have confidently proved why their findings from the main challenge are true.

Some pupils may need help in keeping track of the different dice. A frequent reminder by asking "What were the rules?" may be necessary. Some pupils might also benefit from starting with a simpler version of the task using fewer dice.

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?