Dicey array
This is a game for two players.
You will need two 0-9 dice (or you could use our interactive dice), some counters, pen and paper.
Watch the video below (which has not got any sound). It shows two people playing the game.
Can you work out how to play?
What do you think the rules might be?
If you are unable to view the video, the rules of the game are hidden below.
The aim of the game:
To be the first player to reach the target score.
How to play:
1. Players agree a target score to reach.
2. Player 1 rolls the two dice and finds the total of the two numbers.
3. Player 1 counts out that number of counters and creates as many rectangular arrays as they can using that number of counters. They score a point for each correct array. (Each array must use all the counted-out counters.)
4. Player 2 rolls the dice in the same way and uses that number of counters to create as many rectangular arrays as possible. Again, they score a point for each correct array.
5. Play continues like this with players taking it in turns to roll the two dice. The points scored in each player's second turn are added to the number of points they scored in their first turn to make a running total. Points scored in their third turn are added on again, and so on.
6. The winner is the player who reaches the target number of points first.
Have a go at playing the game several times.
Which dice totals are good to get? Why?
Which dice totals are not so good to get? Why?
Does it matter if you go first or second?
What happens if you can choose to add or subtract the two dice numbers? How does this change the game?
How will you know that you have found all the arrays for that dice total?
You could make a note of the dice total or totals in each game for which you've been able to draw the most arrays.
Why can you draw lots of arrays for some totals but not for others?
Thank you to everybody who sent in their ideas about this game. We had lots of different thoughts about which dice totals would be good to get or not so good to get - have a look at the ideas below and see which you agree with.
To start with, we had lots of solutions sent in from pupils at the ABQ Seeb International School in Oman. Jannah suggested that 12 would be a good total to get since there are many options, and 7 wouldn't be a good total to get since you would only have two choices.
Mohammed said:
The best dice totals are 18 and 20 and 12 because they have the most factors.
Sara said:
The best total to get is 12, 18 because you can make more factors (6).
The dice totals that are not so good to get are 2, 3, 5 and 7 because you get fewer factors (2).
Thank you all for sending in these ideas! The children from Burke Ward Primary School in NSW Australia sent in their wording of the rules and agreed that 12 and 18 were the best totals to get:
The best combinations to roll were 12 and 18 as they are the composite numbers that have the greatest total of combinations (arrays).
Ilham from St Aidan's Catholic Primary Academy in England said:
The best dice total to get is 6 + 6 = 12 since you can make 3 arrays and the worst dice totals include: 2, 3, 5, 7 and 11.
Well done, Ilham - if we are using two 0-9 dice, are there any other dice totals that would be just as bad as the ones you've found?
Most of the pupils who sent in their ideas said that it doesn't matter whether you go first or second because this is a game of chance. However, Taqiya from the ABQ Seeb International School in Oman thought that going first or second does make a difference. Why might it be easier to win if you go first?
Taqiya also had some ideas about how the game would change if you could add or subtract the numbers:
The way this changes the game is because in the game you have to add both the dice numbers and multiply but if you subtract a dice total then your number will decrease and you'll have a less likely chance to win.
Good idea, Taqiya. Why might having a smaller total make it less likely that you'll win?
Jannah thought that being able to add or subtract could be a good thing or a bad thing:
Adding and subtracting would give you more choices or would give you fewer choices. It depends on the number you get.
The children from Burke Ward Primary School found that if you can choose to either add or subtract, this means you have more chance of ending up with a helpful number:
We found that if you gave the person a chance to add or subtract the two dice values, it sometimes meant the difference between a prime or a composite number.
Good idea! Why are prime numbers particularly unhelpful in this game?
Ilham suggested one benefit of being able to subtract:
If you could choose to subtract you could make a more exact number (e.g. 5 - 3 = 2) and you might even step down into negative numbers!
That would certainly make the game very different! I wonder if we could still make arrays using negative numbers?
Naomi from the International School of Brussels in Belgium noticed something interesting about the number of arrays you can make for each dice total. She said:
Through the first game I realised you can only get even numbers because you're able to do the opposite, for example 4x2 and 2x4 (which are the same thing), but for the second game I realised for 3x3 there is no opposite because 3 and 3 are the same number so I got an odd number for how many possibilities there are.
This is very interesting, Naomi. It looks like having a dice total of 9 means you were able to make an odd number of arrays. I wonder which other dice totals would make an odd number of arrays?
Why play this game?
This low threshold high ceiling game offers a meaningful and motivating context in which learners can deepen their understanding of factors, multiples and primes, and develop their fluency with finding factor pairs.
Possible approach
This problem featured in an NRICH Primary webinar in April 2021.
Play the video to the group, simply saying that you'd like them to watch carefully, in silence, to see whether they can work out the rules of the game. The video has no sound and includes three turns altogether, so you might choose to play the whole clip straight away, or you could pause it after player 1's first turn (about 1:20 in).
Invite learners to talk in pairs about the possible rules and how the game is won. Emphasise that they may not be completely sure and that is alright. They may even have some questions to seek clarity. After a suitable length of time, continue to play the video or show it again so that learners can check their initial thoughts.
Bring everyone together again to facilitate a whole group discussion in order to agree on the rules. You could reveal them on screen by clicking the 'Show' button on the problem page. Once everyone is clear, allow them time to play the game several times in pairs without saying much more yourself. It is important that learners are able to 'get into' the game before being expected to analyse it in detail. You may decide that you give children the freedom to develop their own ways of recording their arrays.
You could then invite the group to begin to think about which dice totals are good to get and why (if they haven't done so already). At this point, you could put them in groups of four so that they play two against two. This gives them the opportunity to discuss their thinking with their partner. Watch out for those pairs who have developed a systematic way of working so they are ensuring they find all possible arrays. You may wish to lead a mini plenary to focus on this.
In the final plenary, you could share thoughts about 'good' and 'not so good' dice totals. Encourage learners to articulate their reasoning clearly and suggest that anyone in the class can ask questions to seek clarification. You can ask another member of the group to re-phrase the reason in their own words, which not only helps that individual but also may help other learners who had not followed the reasoning the first time round. Where appropriate, support learners to use correct mathematical vocabulary, for example 'factor', 'multiple', 'prime', or use this as an opportunity to introduce that vocabulary if your class has not met it before.
Key questions
How do you know you have created all the arrays that can be made using that number of counters?
Which dice totals have you found make lots of arrays?
Which dice totals don't make many arrays?
Can you explain why?
Possible support
You may prefer to have pupils play in groups of four from the start, with one pair playing against another pair. That way, they can collaborate as they create their arrays.
Possible extension
Learners could tweak the rules of the game. What would happen if they choose a different target score? What would happen if they use different dice? Encourage learners to ask their own 'what if...?' questions too.