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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# The Dice Train

## THE DICE TRAIN

#### YOUR CHALLENGE

Obeying all the rules, how many solutions are possible?

You can make models like this one or you could make it longer.

Each one you make is to have the funnel on top of the front dice.
### Why do this problem?

This problem challenges pupils in a novel situation, which may seem simple at first. It combines spatial awareness and number awareness.

### Possible approach

## You may also like

### Prompt Cards

### Consecutive Numbers

### Exploring Wild & Wonderful Number Patterns

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

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30 April (Primary), 1 May (Secondary)

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Age 7 to 11

Challenge Level

- Problem
- Student Solutions
- Teachers' Resources

This dice model represents an old blue steam train with a white funnel on the engine at the front. The dice that make up the train are joined using two rules.

RULE 1: Faces that touch each other have the same number.

So, underneath the white dice is a $3$ touching a $3$ on the blue dice.

The blue dice has a $6$ on the face that touches the $6$ on the middle blue dice.

The middle blue dice has a $1$ that touches the $1$ on the last dice.

RULE 2: The number on the top of the funnel must equal the total of the numbers showing on top of the remaining dice (carriages) that can be seen.

So, the $4$ on top of the funnel equals the two $2$'s on top of the blue carriages.

RULE 3: Always use four or more dice - so you have at least two 'carriage numbers' to add up.

RULE 1: Faces that touch each other have the same number.

So, underneath the white dice is a $3$ touching a $3$ on the blue dice.

The blue dice has a $6$ on the face that touches the $6$ on the middle blue dice.

The middle blue dice has a $1$ that touches the $1$ on the last dice.

RULE 2: The number on the top of the funnel must equal the total of the numbers showing on top of the remaining dice (carriages) that can be seen.

So, the $4$ on top of the funnel equals the two $2$'s on top of the blue carriages.

RULE 3: Always use four or more dice - so you have at least two 'carriage numbers' to add up.

Obeying all the rules, how many solutions are possible?

You can make models like this one or you could make it longer.

Each one you make is to have the funnel on top of the front dice.

Introduce dice to the children by asking what they already know about them. You could then talk about what numbers are visible when they are on a table. Next, make the train shape using differently coloured dice and talk about what numbers are on top.

When they are happy with that, introduce the rule about the 'funnel number' and the 'carriages numbers'. They can then talk about the numbers that face each other. When they are comfortable with the rules they can start setting about finding examples that work.

You may wish to stop at a suitable moment to bring the children back together for a short time (a mini plenary). You could ask children to share ways of working which may help all learners in the group progress.

### Key questions

What numbers have you got that equal each other?

Tell me how you are finding more solutions.

When they are happy with that, introduce the rule about the 'funnel number' and the 'carriages numbers'. They can then talk about the numbers that face each other. When they are comfortable with the rules they can start setting about finding examples that work.

You may wish to stop at a suitable moment to bring the children back together for a short time (a mini plenary). You could ask children to share ways of working which may help all learners in the group progress.

Tell me how you are finding more solutions.

These two group activities use mathematical reasoning - one is numerical, one geometric.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.