# Domino Pick

Are these domino games fair? Can you explain why or why not?

Erik and Sanjay have these dominoes laid out:

Image

They pick a domino in turn.

Erik can only choose dominoes which have an odd number of spots in total.

Sanjay chooses the dominoes which have an even number of spots.

How many will they each be able to pick up? Is this fair? Why?

Now they change the rules. They lay out the dominoes again.

Erik can only choose dominoes where the total number of dots is a multiple of $3$.

Sanjay chooses the dominoes where the total number of dots is not a multiple of $3$.

Who is able to pick up the most dominoes now? Is this fair? Explain your answer.

Have you found out the total number of spots on each domino?

How many dominoes would they have each if the game was fair?

How many dominoes would they have each if the game was fair?

Well done Annie (who didn't give her school) and Rukmini from Dowanhill Primary who sent well explained solutions. Rukmini says:

The totals which were odd were: 1, 3, 3, and 5. There were 4 odd-total dominoes. Erik can only choose these 4 dominoes.The totals which were even were: 2, 4, 2, 4, and 6. There were 5
even-total dominoes. Sanjay can only choose these 5 dominoes.

In the next game Erik can only choose a multiple of 3, which are: 6, 3 and 3. So Erik can choose 3 dominoes.

In this game, Erik can choose only 4 and Sanjay can choose 5. Sanjay has
more, and that's not fair.

In the next game Erik can only choose a multiple of 3, which are: 6, 3 and 3. So Erik can choose 3 dominoes.

And Sanjay can choose 6, so it's not fair
again. Sanjay can pick up the most dominoes.

Children from WHS pointed out that it cannot be fair because there are only 9 dominoes and this does not divide equally by two. A very good point, well done.

### Why do this problem?

This problem allows you to introduce ideas of mathematical fairness. It is a useful context in which to focus on children's explanations. It is also a good practice in addition.

### Possible approach

If you have an interactive whiteboard, you may find our Dominoes Environment useful for this problem.

Children often have ideas of fairness based on their real experiences and common sense rather than seeing it from a mathematical view. Lots of discussion may be needed to help them grasp this concept. Having real dominoes available for them to use and physically sort will be helpful. (You could pre-select the set they need for the problem, or include this as an initial task for the
pupils.)

You could introduce this problem by setting up a similar activity with you 'against' the class, using large dominoes or dominoes on the interactive whiteboard. Set criteria that are deliberately biased towards you, such as you can collect all the dominoes which have a total of four or less, the class can choose those which have totals of five or more. If you play a second time, will the class be able to win? Why or why not?

Children could work in pairs on the first part of the problem, recording their ideas on mini-whiteboards, for example. Then it would be useful to bring the whole group together, perhaps on the carpet with some large dominoes, to talk about how they approached the task. Invite a pair to explain what they did, using the large dominoes. Draw attention to those learners who give sound reasons for the fairness of the activity, based on whether or not they each have the same number of dominoes. When children try the second part of the problem, encourage them to articulate their explanations in a similar way.

### Key questions

How many dominoes would Erik have?

How many dominoes would Sanjay have?

Is this fair? Why?

How could you make it fair?

### Possible extension

Can children make up an activity which would be fair? You may decide they should use the same set of dominoes (in which case, as there is an odd number, they will need to make sure there is an odd number of dominoes which neither child can choose), or they could use any from the full set.

### Possible support

It might be helpful to have two sheets of paper which children can label as odd/even or multiple of $3$/not multiple of $3$, so that they can physically group the dominoes and easily remember which is which.