Less is more
Problem
The video below introduces this challenge:
You can have a go using this interactivity:
If you are working away from a computer, you could treat this as a game for two people, or play in two teams of two.
You will need a 1-6 or 0-9 dice. Our dice interactivity can be used to simulate throwing different dice.
Each team should draw some cells that look like the picture below. (The cells on the left don't need to be a different colour, but we will refer to those cells later.) Alternatively, you could print off this sheet of cells.
Version 1
You will need to throw the dice eight times in total. After each throw of the dice, each team decides which of their cells to place that number in.
When all the cells are full, each team will have created four two-digit numbers.
Teams then check if their number sentences are correct:
- If both number sentences are correct, the team scores the sum of the two two-digit numbers written in the blue boxes.
- If only the first number sentence is correct, the team scores the two-digit number in the top blue boxes.
- If only the second number sentence is correct, the team scores the two-digit number in the bottom blue boxes.
The winner is the team with the higher score. See the hint for some examples of scoring.
Have a go at playing the game and keep a running total of your scores.
- Who is the winner after ten rounds?
- Who is the first to reach 500 points?
In between rounds, teams might like to try to find the highest possible score they could have achieved, if they had known the eight numbers in advance. Their new scores can be added to their running totals.
Will this affect your strategy in the next round?
Version 2
Have a go at playing the game in a similar way to Version 1, but this time, note down all eight dice rolls before deciding where to place them.
Keep a running total of your scores.
Who is the winner after ten rounds?
Who is the first to reach 500 points?
Final challenge:
Now, imagine that the numbers 1, 2, 3, 4, 5, 6, 7 and 8 have been thrown.
Where would you place them in order to get the highest possible score?
Can you provide a convincing argument that you have arranged the numbers in the best possible way?
You may like to check whether you have indeed got the maximum score by typing the numbers 1 to 8 (without commas and with no spaces between them) into the 'Values' box in the Settings of the interactivity above, and then testing your solution.
An interesting follow-up to this game is More Less is More, which again challenges you to create correct statements, but this time after carrying out some simple calculations.
This activity featured in an NRICH student webinar in November 2022.
Getting Started
Here are some examples to help you understand the scoring system:
Score = 57 (both number sentences are correct)
Score = 33 (only the bottom number sentence is correct)
With the numbers 1-8, it is possible to score more than 117!
Student Solutions
Thank you to everybody who sent us their thoughts about this game. We received a couple of solutions for Version 1 and a lot of solutions for Version 2!
Version 1
Ilan from Twyford School in the UK considered which strategy worked best with a 1-6 dice:
The key to solving ‘less is more’ is to take a risk and put a large number on the smaller side. Here is an example of the perfect round to win:
65 < 66
65 < 66
This is because the left has to be smaller and it is quite unlikely to roll large numbers so this is quite a risky way of doing it.
This is a less risky round:
45 < 48
39 < 49
The risk to doing the first method is that if you have already put sixes in the first column there is a possibility that you will not get the number you want, in this case sixes and it will not work and you will be deducted a lot of points. In conclusion, I think that going in the middle (not too safe not too risky) so I think the best numbers to put on the right would be 51 or 56 or something along those lines.
Dhruv from St. Anne's RC Primary School in the UK also used a 1-6 dice, and they sent in a picture to explain their method. You can click on the picture to see the full-size solution:
Good ideas, Ilan and Dhruv! It looks like you're both hoping to roll 6s or 5s. I wonder why Dhruv thinks it is best to fill in the tens spaces on the right-hand side first?
Version 2:
We received a lot of solutions from pupils at Halstead Prep School in England, explaining why the highest possible score is 127. Leticia sent in this explanation:
I wrote out out all the numbers from 1 to 8 and then picked 5, 6, 7 and 8. I then arranged them so that I had a 7 and a 5 on the left hand side. Then I put 8 and 6 on the right hand side. I then put the biggest two of the remaining numbers on the left hand side to get 74 and 53. Then I had 81 and 62 on the right hand side and that works. Then I added 74 and 53 and I got 127.
Christina agreed with Leticia's method of considering the largest digits first:
The highest score is 127. You should use 7 on the left and 8 on the right as it uses the less than sign. Then you fill in the units with the leftover lower digits. For the next calculation you would put 5 on the left and 6 on the right for the same reasons. You fill the units with the other numbers. This is how you get the highest score. For the units I put 3 and 4 on the left as they are the highest of the other numbers.
Amelie described a similar method:
The highest number possible is 127. The reason for this is:
You want the highest score on the left. So, use the largest numbers possible. 7 is less than 8 and 5 is less than 6. These are the largest tens digits possible. Next, we move on to the units digits. 1, 2, 3 and 4 remain. It doesn’t matter which numbers you use, as if the tens digit is larger, the number will be larger. To get the highest score possible, we will use 3 and 4. Our result is:
74 < 81
53 < 62
74 + 53 = 127
Our answer is 127.
Well done to all of you for writing out your thinking so clearly. Thank you as well to Lila, Lottie, Ananya, Viva, Charlotte, Evie, Madeline and Emily who all sent in excellent justifications for why 127 is the maximum possible score.
We also received some similar solutions from children at Bishop's Castle Primary School in England. Lewis and Ryan sent in this explanation:
I put the highest number (8) in the top right tens column and the second highest number (7) in the opposite tens column because I knew I needed the largest total on the left hand side. The next highest number (6) I put in the tens column on the right hand side in the row below and the next highest number (5) in the tens on the left hand side. I took a similar approach with the position of the ones. This gives a top score of 127.
Lucy and Fynn had a similar strategy:
The 7 is smaller than 8 therefore you can use them on the same row in the tens. This strategy works for 6 and 5 as well. You can put 4 and 3 next to the 7 and 5 either way round in the units to make the largest possible correct numbers on the left hand side. The same strategy works for 1 and 2 on the right hand side. The highest score with digits 1-8 was 127.
Sid from Twyford School explained how they solved this problem by starting with the highest digits:
If you have the numbers 1,2,3,4,5,6,7 and 8, the way to get the best score would be looking at the highest numbers and going down to the lowest numbers. First you look at the eight, the eight cannot be in the first column in the left section as nothing else is bigger than eight. What you could then do is put 8 in the first column in the right section and seven in the first left section. That then means you now have the numbers 1,2,3,4,5 and 6. Like the 8 you can't put the 6 in the first left section so you would want to put it on the right first section under the 8, then you can have the number 5 in the left first one under the 7, you can then put 4 and 3 in the last places on the left and 2 and 1 on the right then your total would be 127.
Dhruv sent in this picture explaining their method:
Thank you all for sending in these solutions!
Teachers' Resources
Less Is More
This is a game for two people, but is better played in two teams of two.
You will need a 1-6 or 0-9 dice. The interactivity here can be used to simulate throwing different dice.
Each team should draw some cells that look like the picture below. (The cells on the left don't need to be a different colour, but we will refer to those cells later.) Alternatively, you could print off this sheet of cells.
Version 1
You will need to throw the dice eight times in total. After each throw of the dice, each team decides which of their cells to place that number in.
When all the cells are full, each team will have created four two-digit numbers.
Teams then check if their number sentences are correct:
- If both number sentences are correct, the team scores the sum of the two two-digit numbers written in the blue boxes.
- If only the first number sentence is correct, the team scores the two-digit number in the top blue boxes.
- If only the second number sentence is correct, the team scores the two-digit number in the bottom blue boxes.
The winner is the team with higher score. See the hint for some examples of scoring.
Have a go at playing the game and keep a running total of your scores.
- Who is the winner after ten rounds?
- Who is the first to reach 500 points?
In between rounds, teams might like to find the highest possible score they could have achieved, if they had known the eight numbers in advance. These scores can be added to their running totals.
Will this affect your strategy in the next round?
Version 2
Now, imagine that the numbers 1, 2, 3, 4, 5, 6, 7 and 8 have been thrown.
Where would you place them in order to get the highest possible score?
Can you provide a convincing argument that you have arranged the numbers in the best possible way?
Have a go at playing the game just as you did in Version 1, but this time note down all eight dice rolls before deciding where to place them.
You can score in the same way as you did in Version 1. This time it could be the first to reach 1000 points who wins!
An interesting follow-up to this game is More Less is More, which again challenges you to create correct statements, but this time after carrying out some simple calculations.
Why play this game?
This game is thought-provoking and very engaging. It encourages discussion of place value, alongside valuable strategic mathematical thinking, and it helps learners become more familiar with the mathematical symbols for 'greater than' and 'less than'.
Setting up the game with students working collaboratively with a partner offers the chance for them to focus on all five key ingredients that characterise successful mathematicians.
Possible approach
Before the interactivity was added, this problem featured in an NRICH Primary and Secondary webinar in November 2022.
You can introduce the game by playing the video, or recreating the video contents for yourself using the interactivity.
Invite students to have a go at the game in pairs, ideally using the interactivity on a computer or tablet. Set them the challenge to try to find the highest possible scores in fewer than three attempts.
Once they have had chance to play several rounds, bring everyone together to discuss their thinking. How are they deciding where to place the numbers? How are they trying to maximise their score?
Having shared strategies, students might enjoy playing in teams of two against each other. Who can reach 500 points first? Or who can make the highest score after five rounds?
The first challenge of More Less is More (called 'sum-sum') is a natural extension to this task.
Key questions
How are you trying to make sure each number sentence is true, while still managing to get a high score?
How are you deciding which cells to fill in first?
Possible extension
An interesting follow-up to this game is More Less is More, which again challenges learners to create correct statements, but this time after carrying out some simple calculations.
Possible support
Learners could try a single-digit version of the game initially, rolling the dice four times.
Learners could be provided with number cards that they can move around the grid to consider different options.