# 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.

Well done to everybody who had a go at this problem. Mitch from Australia sent in this picture and explanation:

Hi,

This is my picture for the dice stairs.

I started by building the third stair so the top was 4. Then I made the first stair 2. Then I worked on the second stair to make 3 by rotating the dice.

Good ideas, Mitch! Interestingly, it looks like Mitch has used a mixture of left-handed dice and right-handed dice to build these stairs, but the solution using all one type of dice would be very similar.

Maryam, Zaina, Nasra and Aagesh from Stag Lane Junior School in England also found the solution with 2, 3 and 4 on the top for Challenge 1. They said:

We found 432 on Challenge 1:

For the top numbers 1 on the right side of the two, two 1s on the left side of the 3 tower and two 6s on the right side of it, three 6s on the 4 tower and three 1s on the right side.

Challenge 2:

For this challenge we only add one last tower and the number on the top was 5 the left side was all 1s and the right all 6s.

These are some very clear descriptions - I can see that this would make the tower with 2, 3, 4 and 5 on the top by using 1s and 6s to stick each tower together.

They said:

We did go on to explore 10 dice and 15 and decided any set that were starts (involving 1) or ends (involving 6) would not work.

This is very clearly explained! Thank you all for sharing your ideas with us.

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.)