# Roll these Dice

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

## Problem

*Printable NRICH Roadshow resource*

In this activity, you will need two dice that are the same and one that is different. We have chosen two red dice and one green dice.

We are going to roll the dice and add up the numbers on the two red dice, then subtract the number on the green dice.

In the picture above, the number on one red dice is $4$ and the number on the other red dice is $5$, with the number on the green dice being $3$. We add together $4$ and $5$ to make $9$ and then subtract the $3$ which gives us a final answer of $6$.

You'll need to roll your dice many times and see what numbers you make each time after doing the addition and subtraction.

It would be good to find out:

- what are the final answers by doing the addition and subtraction each time?
- what are all the different possible numbers?
- is there a good way of making sure you find them all?
- how will you record what you've found out?

Now have a go!

Then you could ask yourself, "I wonder what would happen if, instead, I ...?''

## Getting Started

What are the possible results of adding the red dice?

Is there a good way of making sure you find all the possibilities?

What are the possible results of taking away the number on the green dice from the red totals?

How will you record what you've found out?

## Student Solutions

Molly from Churchers College Junior School sent in these well thought out ideas about this activity.

* I have found out that all the different possible answers are between $ -4$ and $11$ including $11$ and $-4$. It is not possible to get any answers over $11$ and below $-4$.

I recorded all of this data in a table.

* I have found out that if all $3$ dice are the same the total will become the value of one of the die ( Eg. $1+1=2-1=1$ ).

* this would be the same conclusion as above if any $2$ of the die were the same. (E.g. $2+1=3-1=2$)

* I know there is a $50/50$ chance of the answer being odd or even because;

odd + odd - odd = odd

even + even - even = even

odd + even - odd = even

odd + odd - even = even

even + even - odd = odd

odd + even - even = odd

These are all the posible ways of adding the dice.

Thank you for reading my solution I hope all is correct.

Molly ;-)

Indeed Molly it is very good and I am impressed that you did this and came to those conclusions. You could of course extend the exploration by wondering about using $4$ dice and deciding whether to subtract just $1$ of those or maybe $2$.

Ben, Harry, Will and Lucas from Tarporley Church of England School also worked on this activity and this was their report:

There are four of us so two of us wrote ALL of the combinations down [one from $6+6-1$ and one from $1+1-6$]. There were $216$ possible calculations. At the same time the other two of us worked out which is the most likely answer[which is $4$]. Once we did that we were done.

Sion from the same school added this extra piece of information;

There are $225$ ways and your answer is the numbers $3$ and $4$. By finding all the $225$ calculations you then make a tally chart to show the most popular number. Finally you count up the number and then your answer should be $3$ and $4$.

We also had a number of good ideas from North Molton, namely, Michael, Jack, Beth, James and Sam.

Bram”¨ from the British School of Bucharest”¨ in Romania”¨, sent in what I think is the first from Romania, - well done and thanks - saying;

There is a higher probability to get $6$ than $2$ eg. there are fewer ways to get $2$ because there are $13$:

$1+2-1=2$ , $1+3-2=2$ , $1+4-3=2$ , $1+5-4=2$ , $1+6-5=2$ , $2+3-3=2$ , $2+4-4=2$, $2+5-5=2$ , $2+6-6=2$ , $3+3-4=2$ , $3+4-5=2$ , $3+5-6=2$ , $4+4-6=2$

and for $6$ there are:

$1+6-1=6$ , $2+5-1=6$ , $2+6-2=6$ , $3+4-1=6$ , $3+5-2=6$

Thanks you all, a great effort.

## Teachers' Resources

**Why do this problem?**

This activity offers practice in addition and subtraction, including negative results, but the main aim is for pupils to concentrate on making sure that all the ways of rolling the three dice are reached. This will need some sort of system and you could focus on how answers could be recorded.

### Possible approach

You could introduce the problem using real dice and modelling the calculation a few times so that pupils get a feel for it. Once a few results have been recorded on the board, invite the pupils to speculate on how many different results there might be.

Ask them to work in pairs or small groups on the problem, saying very little else at this stage, but after a short time, bring them together again to share insights so far. Discuss the range of answers that learners have found up to that point and make sure they are happy with subtracting one number from a smaller number. (Using a number line which includes negative numbers might be helpful at this point.) Invite some pairs to describe how they are working. Some may be throwing real dice, others may be listing numbers. Encourage some sort of system so that they can be sure no results are left out. You could ask children to suggest ways of recording which would help - this could be in the form of a table or chart, but allow pupils to choose a way that suits them.

### Key questions

What are all the different possible numbers?

What are the final answers by doing the addition and subtraction each time?

Is there a good way of making sure you find all the possibilities?

How will you record what you've found out?

### Possible extension

### For more extension work

Pupils can be challenged to use multiplication as well as addition and subtraction. After some experimentation, they could try to predict totals that will NOT be possible and explain their predictions.

### Possible support

Most children will need dice to begin this activity. Some pupils will benefit from an adult working alongside them and asking questions along each step of the way until their confidence has increased.