Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Six Ten Total

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

**The main challenge**

#### Taking it further

#### Final challenge

## You may also like

### Pebbles

### Bracelets

### Sweets in a Box

Or search by topic

Age 7 to 11

Challenge Level

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.*

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?