Double or Halve?
Problem
This is a game for two players.
You will need a dice (1-6 or 0-9), or you could use our interactive dice.
How to play:
- Decide on a target number. This is the total that both players are trying to make.
- Player 1 throws the dice. They can choose whether to double the number shown or halve the number shown.
- Player 2 throws the dice. In the same way, they can choose whether to double the number shown or halve the number shown. Player 2 adds their number onto Player 1's number to make a running total.
- Play continues like this with each player rolling the dice, halving or doubling the number, and adding the result onto the running total.
- The winner is the player who reaches the agreed target exactly.
Here are some questions to think about:
Must each player always take a turn?
Does it matter if you go first or second?
Are there any particularly good numbers to choose as your target? Why or why not?
What are you thinking about in order to try to win the game?
Student Solutions
We haven't yet received many solutions to this activity. If you notice anything interesting when you play this game, please email us with your ideas.
Joshua from Spring Hill Primary School in Australia had these ideas:
Q1: Does it matter if you go first or second? Why or why not?
It does not change any chance of winning.
Q2: Are there any particularly good numbers to choose as your target?
There are no specific numbers that are better to choose, it has the same likelihood.
Q3: What are you thinking about in order to try to win the game?
When you are close to the target number you should always half unless doubling can win it.
Good ideas, Joshua! I wonder why it might not matter if you go first or second?
Hannah from Oyster Bay Public School in Australia used a similar strategy:
I played this with my mum and these are some of the strategies that we did:
- We chose the number 82. I rolled first and doubled it.
- We both kept doubling our numbers to get to the number 82.
- We wrote the number rolled and the number doubled to keep track.
- We added each other's numbers.
- When we came close to 82 we started to half to make our number smaller.
- We skipped turns until we came up with a correct number that brought our number to our target number of 82.
- I won!
Well done Hannah! Thank you both for sending in your ideas.
Teachers' Resources
Double or Halve?
This is a game for two players.
You will need a dice (1-6 or 0-9), or you could use our interactive dice.
How to play:
- Decide on a target number. This is the total that both players are trying to make.
- Player 1 throws the dice. They can choose whether to double the number shown or halve the number shown.
- Player 2 throws the dice. In the same way, they can choose whether to double the number shown or halve the number shown. Player 2 adds their number onto Player 1's number to make a running total.
- Play continues like this with each player rolling the dice, halving or doubling the number, and adding the result onto the running total.
- The winner is the player who reaches the agreed target exactly.
Here are some questions to think about:
Must each player always take a turn?
Does it matter if you go first or second?
Are there any particularly good numbers to choose as your target? Why or why not?
What are you thinking about in order to try to win the game?
Why do this activity?
This activity provides a context for doubling and halving, as well as adding to keep a running total. The children will also be thinking about strategies they could use to win the game.
Possible approach
This problem featured in an NRICH Primary webinar in June 2021.
At the start of the lesson, model the first few turns of the game on the whiteboard. If you have another adult in your classroom, the two of you could play against each other. Alternatively, you might play against two children who will need to be briefed beforehand! At this point, do not tell the rest of the class the rules of the game, instead invite them to watch the game being played
and consider what the rules might be.
Ask chldren to share their noticings and their suggestions about the rules, and use the ensuing discussion to clarify how to play.
Once the rules have been established, play again, this time involving the whole class either against you, or in two teams. Ideally, all children will have access to practical equipment that they could use to help them double/halve. Each time the dice is rolled, allow time for learners to double and halve that number, perhaps recording on a mini whiteboard, before asking what number
they will add to the running total.
This will provide a good opportunity for discussing what happens when an odd number is rolled, as halving this will not give a whole number answer. Invite children to suggest what might be done in these cases. Is halving still 'allowed' in this game if you roll an odd number?
Once the pupils have had this experience of playing the game altogether, they can play in pairs, writing down their target number and running total on a whiteboard or in their books for each game.
Stop the class halfway through the session to discuss what strategies they are using. When do children think it is better to halve rather than double their number? Talk about whether pupils need to think harder at the beginning or near the end of a game, and encourage pupils to think about what strategies they need to use at the end of the game in order to win. Are they allowing a player to
'pass'? If not, what happens if a player goes over the total?
Key questions
What is double the number you rolled?
If we halve the number you rolled, will the answer be a whole number? How do you know?
Why did you choose to double/halve that number?
What will happen if you choose a different target?
Possible extensions
Extension 1: Pupils could use a 0-9 dice rather than a 1-6 dice, which will lead to them doubling and halving slightly bigger numbers. They could also be encouraged to choose a larger target number.
Extension 2: If pupils are very confident with the concept of halving, introduce them to what happens when an odd number is halved. Encourage them to find their own ways of recording this - they might like to draw a simple shape and shade half of it, or write 'two and a half' in words.
Possible support
Numicon, multilink cubes or similar may help children to calculate the halves and doubles.