Make 37
2 out of 3
Four bags contain 1s, 3s, 5s and 7s. Can you pick ten numbers from the bags so that their total is 37?
Problem
Four bags contain a large number of 1s, 3s, 5s and 7s.
Can you pick ten numbers from the bags that add up to 37?
What Numbers Can We Make? is a suitable follow-up problem.
Getting Started
What numbers can you make? Do they have anything in common?
Have you made $37$ with a different number of numbers? How many? Do these numbers have anything in common?
What do you notice about the numbers in the bags?
Student Solutions
Well done to everybody who had a go at this task. Lots of children worked out that it was impossible - you can't make 37 with a combination of ten 1s, 3s, 5s and 7s. Some children explained why they thought this was impossible, and we've focused on those solutions here.
Luca from The Village School in the USA said:
You can't get 37 by adding together 10 odd numbers.
Freddie explained why this isn't possible:
I think that it's impossible because you cannot make an odd number with an even number of odd numbers.
Well spotted! I wonder why we can't make an odd number with an even number of odd numbers?
Ethan from Hong Kong International School also explained why this is impossible:
There is no solution to this problem, since there are only odd numbers. And we know that odd + odd is always even. And there are ten numbers which makes it 5 even numbers, and even + even is always even. However, 37 is odd.
Thinking about adding five even numbers instead of ten odd numbers makes it much more obvious that the answer will always be even. Ci Hui Minh Ngoc from Kelvin Grove State College in Brisbane, Australia sent in this picture to represent adding the ten odd numbers together, starting by adding pairs of odd numbers to make even numbers:
Can you see how they've added the numbers at each step of the table?
Ryota, Chisato, Dahlia and Isabelle from The Perse School in Singapore also agreed that it is impossible to make an odd number (like 37) by adding together ten odd numbers. Ryota sent this explanation, which includes very clear ways of thinking about odd and even numbers.
Maisy and Will from Pencoys Primary in the UK found out that there wasn't a solution by working systematically:
Over a series of lessons, we used trial and error to solve the problem. We were so close and kept either reaching 36 or 38! This made us work systematically and we worked through all of the possibilities there are. Now more determined than ever, a whole lunch time later, we finally found our theory:
When you add odd numbers such as 1s, 3s, 5s and 7s used in ‘Make 37’, if you add them an even amount of times (such as 10) you will always get an even answer e.g. 7+3+1+7+5+1+7+5+1+1=38 which does not work. This proves our theory right as the only options to select were odd numbers. Another example 5+5+5+5+7+3+3+1+1+1=36.
If it were 9 numbers or 11 numbers it would be possible.
It sounds like you worked really hard on this, and then you were able to show that it would never be possible with ten numbers - well done! Ellen from Cheam Common Junior Academy in the UK had a similar idea:
I started off by trying to add ten numbers to make 37. After several tries, I noticed that I could make 36 or 38, but not 37 itself. I noticed that the numbers 1, 3, 5 and 7 are all odd numbers. I experimented with adding groups of odd numbers together and we found out that when adding two odd numbers together, the answer was always even e.g. 7+7=14 (even). When I added three odd numbers together the answers were always odd e.g. 9+9+9=27 (odd) 5+5+5=15 (odd). I noticed that if I added an even number of odd numbers together, then the answer would always be even. So it is impossible to make 37 from ten odd numbers!
Ryan and Zac from Banstead Prep School in England used multiplication instead of addition to show that the problem was impossible:
In our solution to this problem, we started by writing out all of the numbers and gradually crossing them off after every time we used.
The closest that we got was 36 (ten numbers) 38 (ten numbers) and 37 (nine numbers). After a long while we started questioning even and odd multiplication and addition.
Fortunately for us, we remembered that even multiplied by odd = even.
As soon as we had remembered that we found out that it was impossible to get 37 because 10 x odd = even.
Well remembered! I wonder how you could prove that an even number multiplied by an odd number will always be an even number?
We were also sent in similar solutions by: Om from St Leonards Exeter; Hwan from Hong Kong International School; Drayvian and Avi from Banstead Prep School in England; Jade and Aala from North London Collegiate Junior School in the UK; Sophie from Glenfall in the UK; Elliott from Richmond Methodist Primary School in the UK; and Matthew from Devonshire House in England. Thank you all for sending in your ideas.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
Make 37 offers learners the chance to apply basic number properties and operations, and will challenge the idea of mathematical problems having one right answer. It is an opportunity to encourage children to generalise and, even at this level, they can be asked to prove these generalisations.
Possible approach
One way to start is to ask pupils to make 16 using any four of the numbers. Make sure they have paper or mini whiteboards for working on. They will quickly find a variety of answers.
Declare this is too easy and set them on the task of making 37 using ten of the numbers. Then give the pupils time to "play", trying out different numbers for themselves. It would be a good idea to encourage them to check their results with a neighbour! Refrain from asking if it is possible; wait until a pupil suggests it can't be done, to which the response could be: "Why can't it be done?"
It might be appropriate to ask pupils what totals they have been able to make and what they notice about these totals. Turning pupils' attention to the properties of the totals that are possible will encourage them to suggest conclusions. Have they noticed that all the totals are even? This may alert them to the fact that making 37 is much harder than they thought or does not seem to be possible.
Having given the group more time to work in pairs or small groups, draw their ideas together using the key questions below. Once they have drawn conclusions based on odd and even numbers, challenge them to convince you this is the case. Drawing diagrams of odd and even numbers may help pupils to visualise the mathematics.
A follow-up question along the lines of "How could you change the problem so that we can make 37 from ten numbers?" or "How can we change the problem so that it is possible?" provides a useful opportunity to gauge how fully the children have understood the ideas.
Key questions
What do you notice about the numbers in the bags? What do they have in common?
What totals have you managed to make using ten numbers? Do they have anything in common?
Can you make 37 with a different number of numbers? How many?
Can you explain why?
Possible extension
This problem could be used as a starter before tackling What Numbers Can We Make? and the more challenging Take Three from Five
Possible support
You can use a lower total and fewer numbers to make the arithmetic easier without losing the mathematical thinking and reasoning. For example, the challenge could be to make 15 using four numbers.