# Inky Cube

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

## Problem

It is rolled over by tipping it very carefully (with no sliding) along one of the edges so that the next inky mark is right by its side. This gives;

It can roll both left and right and up and down.

When it has done several rolls, without taking it off, the paper looks like this;

Your challenge is to find out:

- where the inky spots are on each of the six faces
- where the cube was first placed
- the route that the cube has taken.

Next, you might like to try this colour version where you have to find the colours for each patch on each of the faces of the cube, as well as the route it took.

Here is a printable version of the marks left by this coloured cube.

## Getting Started

What do you know about the faces of the cube?

How will you remember what you have done so far?

How will you try to find a route that works?

How do you know you have found all the possible routes?

How will you check that your route/s definitely work/s?

Don't forget that each face of the cube is different.

## Student Solutions

Demi, Hannah, Izobel, Celia, Joseph and Michael from All Saints C of E Junior School sent us their solution to this challenge. They said:

First we divided the grid into 12 compartments each of 9 squares (3x3).

Then we covered a large dice in paper and stuck it with sellotape.

Next we drew the top left corner pattern onto a side of the dice.

Then we rolled it down once, then drew on that pattern.

After doing lots of trial and error we found out a route:

We started at the top left corner and then went down 1 space, right 1, and then up 1.

Next we went to the right and down 3.

Then we went left 2, up 1 and 1 to the right! (the finishing point!)

I've drawn a rough sketch to show the route they describe:

They also sent in a net of the cube:

Niharika from Leicester High Schools for Girls told us:

I saw the cube turn in the air. It was hard but I enjoyed it.

Mathematicians might call that 'visualising'. Niharika sent in another solution which is the reverse of the route above.

Niharika also tackled the second part of the challenge which involved a cube with coloured squares painted on it. Firstly, she labelled the grid:

She then went on to describe the route:

(1, 1) --- (2, 1) --- (3, 1) --- (3, 2) --- (2, 2) --- (2, 3) --- (3, 3) --- (3, 4) --- (2, 4) --- (1, 4) --- (1, 3) --- (1, 2)

Niharika explained that she thought carefully about the symmetry of each of the faces and how an odd or even number of 'tips' might affect each face. Fantastic!

Will the reverse of Niharika's route work too, do you think?

## Teachers' Resources

**Why do this problem?**

### Possible approach

### Key questions

### Possible extension

### Possible support