### Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

### Cuisenaire Rods

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

# Dice Stairs

## Dice Stairs

This challenge involves making a 'dice stair' with three steps, as in the picture:

This is how you make a dice stair:

1. Take six dice.
2. Place the dice in three towers: a tower of one, a tower of two and a tower of three.
Each pair of dice faces that touch will only 'stick' if they are matching numbers.
3. Place the towers in the staircase format.
Each pair of dice faces that now touch will only 'stick' if they are matching numbers.
4. The top faces of the dice that make the 'steps' must now show three consecutive numbers in ascending order for this arrangement to be a dice stair.
For example, in the picture above, the top face of the blue dice and the bottom face of the red dice both have matching 3s. The left hand side of the blue dice and the right hand side of the green dice have matching 6s and the bottom of the blue dice has a 4 that matches the top of the white dice. Similar matching is true for all the other dice faces that touch.

CHALLENGE ONE - three steps and six dice
• Make some sets of dice stairs, as described above.
• What do you notice as you explore how to make dice stairs?
• Explain how you found your dice stairs.
• How do you know that you have found them all?

CHALLENGE TWO - four steps and ten dice
• Try the challenge above using four more dice to make dice stairs.
• How have you used what you learnt using six dice?
• How do you know that you have found all the dice stairs arrangements?

CHALLENGE THREE - beyond four steps

What insights do you have that would help you to try to make dice stairs that have five steps?

Justify whatever conclusions you come to when you set out to make your five-step stairs.

### Why do this problem?

This activity was designed for the 2015 Young Mathematicians' Award so it might be a particularly useful activity for a small group of your highest-attaining pupils to work on. It is a useful vehicle for developing systematic approaches. It can be used as an activity to encourage children to explain in written or spoken words what it is they have done.

### Possible approach

Since this activity is aimed at the most confident mathematicians, there will not be much that you have to do to introduce it apart, perhaps, from letting the pupils declare their different ways of answering the first challenge.

Encourage them to discuss their thinking and reasoning as they progress through the challenges.

### Key questions

Tell me about how you are getting a solution for the challenge you are working on.
(The main thing when encouraging the pupils to use their skills is to avoid saying things about what you notice and directing them in your way of attempting a solution.)

### Possible extension

Learners might like to try the other two challenges that were part of the Young Mathematicians' Award 2015: Open Boxes and Centred Squares.