Dice in a Corner
This activity has been specially created for the most able. The intention is that these pupils sort it all out for themselves, with little teacher explanation. Hence these Teacher Notes are shorter than usual.
Dice in a Corner
There are three dice sitting in the corner with the simple rule that where two dice meet there must be the same numbers facing each other.
So, in the first picture above there are $3$'s at the bottom of the red dice and on the top of the middle green and there are $4$'s on the bottom of the green dice and the top of the white dice. The numbers on the seven faces that can be seen are then added and make $21$.
In the second picture above there are $4$'s at the left of the red dice and on the right of the green dice and there are $3$'s on the left of the green dice and the right of the white dice. The numbers on the seven faces that can be seen are then added and make $23$.
Your challenge is to arrange dice (using at least $2$ and up to as many as you like) in a line from the corner, so as the faces you can see add up to $18$ (instead of the $21$ and $23$ above), in as many ways as possible.
Each line of dice must be along or up a wall (or two walls). A line going up is counted the same as a line going along. Remember the dice must touch face to face and have the same numbers touching. The lines of dice must be of a single thickness, so this one below is not allowed;
Why do this problem?
This
activity engages the pupils in both a spatial and numerical context and gives them the freedom to choose how they go about it - visualising in their head, using dice that they have asked for/found and/or making use of a spreadsheet. They can learn a lot from adopting one method and then realising that an alternative method would be
better.
Possible approach
Presented as on the problem page.
Key questions
Open questions such as, "tell me about this . . . . " and
"how did you decide on this approach to finding all the possibilities?"
Possible extension
Using $4$ dice what is the range of different totals you can get?
How many ways are there for the most popular total?
Can you use a spreadsheet to get the results by using formulae rather than just entering data?