# Rolling that cube

My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?

## Problem

We have a dice with raised spots, which have been pressed on an ink pad.

We start by printing one number.

Can you see how the numbers below have been printed by rolling the dice in the four different directions?

We start by printing one number.

Image

Can you see how the numbers below have been printed by rolling the dice in the four different directions?

Image

The dice has been rolled around without sliding, and it has printed the picture below. Your challenge is to find where the cube was placed to begin with and map out the route it has travelled. You may find it useful to print off a copy of the picture on this sheet.

There are two different ways of arranging the numbers on the faces of dice. If you look at the faces '1', '2' and '3' on a dice, you can see that they share a vertex. In a left-handed dice, the '1', '2' and '3' run clockwise, and in a right-handed dice they go anti-clockwise. See Right or Left? for more information about this. If you're using a dice for this activity, first work out which type of dice you have and then use the matching print below.

Right-handed dice print:

Image

Left-handed dice print:

Image

*NB You may find that the six and two are in a different orientation when you try this out with your own right-handed or left-handed dice.*

## Getting Started

How will you know what you have tried so far?

Are there any rolls that you know are impossible?

Are there any rolls that you know are impossible?

## Student Solutions

This activity produced a few replies. Oliver from St. Anthony's sent in;

R R D L L D D R R U L

Tessa, Sally and Kensa from Sherwood State School in Australia sent in their solution like this;

$1, 4, 6, 2, 4, 5, 1, 2, 4, 5, 1, 4$

Hanako and Emilia at Vale Junior School, Guernsey sent in this word document;

First we decided to make a cube to physically test our theories and ideas.

We spotted that there were two impossible routes. These were: the $4$s down the middle and the $1, 4, 1$ combination going across.

These are impossible because you can't have two $4$s next to each other as there is only one four on the dice. The other is impossible as to get from $1$ to $4$, and then to $1$ again, you would have to double back on yourself.

Next, we had to think of a route that bypassed these two impossible combinations. We thought that we could start our route with the $4$ in the impossible $1, 4, 1$ combination so that we didn't have to complete the whole impossible combination.

Image

We checked that our theory was correct by rolling our cube along the grid. As we rolled it, we wrote the next number in the grid as a reflection on the next face of the cube. We tried starting at the top $1$ and the centre $4$ and we found that this route works both ways.

Thank you Hanaho and Emilia for explaining how you did it and what your thoughts were and well done all of you!

## Teachers' Resources

### Why do this problem?

This activity is useful for developing good spatial awareness and offers opportunities for sharing different ways of approaching it.

### Possible approach

Depending on the pupils' experiences, it may be appropriate to start with them all together with a practical example. Using a large dice, they could observe what happens as it rolls and try to predict what face will be at the bottom each time. Show children the picture of the prints which go in four different directions from the centre number, and encourage them to describe what has
happened. Can they roll the large dice to show how these prints have been made?

You could then present the challenge itself and give children time to work independently or in pairs. A copy of the route can be found on this sheet (word, pdf). Try not to direct the way they work as you may be suprised by the methods they choose.

The plenary can then focus on their different approaches. Allow time for all the different ways to be explained and then encourage pairs/small groups to discuss which method they might use if they were presented with a similar problem. Can they justify their choice?

You may be surprised by those learners who find this task challenging and need extra support and those who don't.

You may be surprised by those learners who find this task challenging and need extra support and those who don't.

### Key questions

When working with the simple cube above:

What's happening here?

What can you tell me about the one at the bottom when I roll it this way?

When working on the actual challenge:

How are you working this out?

Is there more than one way of getting there?

### Possible extension

When pupils have managed this activity in a confident way they may like to have a look at Inky Cube which is similar but much harder. You could also give the pupils opportunities to create their own cubes and set challenges for each other.

### Possible support

Some pupils will need practice in carefully rolling the dice.