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?
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:
Left-handed dice print:
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?
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.
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
Rolling That Cube
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?
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:
Left-handed dice print:
Why do this problem?
Possible approach
You may be surprised by those learners who find this task challenging and need extra support and those who don't.
Key questions
Possible extension
Possible support