More Less is More
Problem
More Less is More printable sheet - game instructions
More Less is More printable sheet - blank grids
These challenges follow on from Less is More.
This video below introduces these challenges:
You can have a go at the four different versions 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 pictures below.
In Version 1, you place the numbers after each throw of the dice.
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 be able to check if their number sentence is correct.
In all cases, you score if the sentence is correct. The score is the result of the calculation on the left of the inequality sign. See the hint for some examples of scoring.
The winner is the team with the higher score.
In between rounds, teams might try to find the highest possible score they could have achieved, if they had known the eight numbers in advance. Their new scores could be added to their running totals.
In 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?
Sum-sum
Take-take
Take-sum
Sum-take
Final challenge:
Imagine that you have thrown the numbers 1-8.
What is the highest possible score for each of the games above?
Can you provide a convincing argument that you have got the highest possible score?
A clue is given in the hints.
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.
Getting Started
Sum-sum
Score = 42 + 16 = 58
With 1-8, the maximum score is more than 130 points
Take-take
Score = 55 - 35 = 20
With 1-8, the maximum score is more than 55 points
Take-sum
Score = 42 - 16 = 26
With 1-8, the maximum score is more than 70 points
Sum-take
Score = 12 + 24 = 36
With 1-8, the maximum score is more than 70 points
Student Solutions
Sum-sum
Gerard from Frederick Irwin Anglican School in Australia inserted numbers that make the inequality true:
32+36 is less than 65+26 this is because 32+36=68 while 65+26=91
Gerard's score is 68.
Leticia and Amelie from Halstead Prep School and Rishaan, Swarnim and Eshaan from Ganit Kreeda in India worked out how to get the highest score using the numbers 1 to 8. This is Leticia's work:
I got the four highest numbers 5, 6, 7 and 8. I then arranged them do that I had 5 and 8 on the left hand side and 6 and 7 on the right hand side. I then had the other four numbers and I then figured out that I could do 1+3=4 and 4+2=6 and you can’t have 3+2 and 1+4 because these both add up to five so they are equal. You can’t do 2+1 on the left hand side because that is not the highest combination. So I then did 51 and 83 on the left and 64 and 72. If you did this the other way round the inequality would be incorrect. Then I added 51 and 83 and then I got 134 which is the highest score.
Gowri from Ganit Kreeda explained how to maximise your score given any 8 numbers:
Take-take
Gerard inserted numbers that make the inequality true:
54-56 is less than 44-26 as 54-56= -2
Gerard's numbers give a negative score of -2.
Eshaan, Rishaan, Samaira, Gowri, Vibha, Viha, Vansh, Vraj, Arya, Swarnim, Rudraraj and Renah from Ganit Kreeda worked together to find the best score possible using the numbers 1 to 8:
Swarnim, Vansh and Eshaan had used the strategy of greatest number – smallest number to get the maximum difference.
Rishaan tried the numbers at tens place such as the difference will be maximum and equal for both the sides.
As 8-2=7-1 = 6
He used 8_ - 2_ < 7 _ - 1_
The kids then used the remaining digits at ones place. Again, they used the same strategy as used in the first challenge. They tried to get the difference of 1.
85 – 24 < 76 – 13 or 61 < 63
The Highest Score is 61.
Amelie used very similar reasoning, expressed in a different way:
Take-sum
Gerard inserted numbers that make the inequality true:
31-45 is a negative while 34+14 is positive
Again, Gerard's score is negative (this time -14).
Amelie combined the numbers 1 to 8 to create a higher score:
Note that Amelie probably meant 87-13 rather than 87-14.
The students from Ganit Kreeda managed to get a slightly higher score:
87-12 will give the maximum score we can get on the left side which is 75 and now to get a number bigger than this on the right side we can use the leftover numbers.
87 - 12 < 35 + 46
Sum-take
Gerard inserted numbers that make the inequality almost true:
12+11 is less than 46-23
In fact, now both sides are equal to 23, so this is not actually true!
Amelie used the numbers 1 to 8 to create a high score:
The students from Ganit Kreeda also got a score of 71:
First, we tried to use maximum difference which is 75 on the right side.
But then using remaining digits, the smallest possible addition was 35+46 which is 81, and 81>75.
So, we tried to make right side 74 by changing 12 to 13 as 87-13.
With remaining digits we got 26+45=71 and 87–13=74 OR 46+25<87-13
The Highest Score is 71.
It is actually possible to get a score of 72. Can you see how?
Teachers' Resources
This problem follows on from Less is More.
See the Teachers' Resources of Less Is More for guidance on how this problem could be used in the classroom.
This problem featured in the NRICH Primary and Secondary webinar in November 2022, and in an NRICH student webinar in the same month.