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Make 37

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2
We had quite a few submissions for the challenge. Some noted that it was impossible, but we were looking for rather more - an explanation as to why it was not possible and ways of proving that. Here are some that tried to explain why it was impossible.

Imogen, Molly, Hannah, Charlotte, Elodie and Jess from Sandbach High School wrote:

We quickly got 36 and 38. Only when we looked at the other possible totals from 10 to 70 did we spot the answers were always even numbers! This puzzle is impossible to prove!  As an extension we looked at making 37 with different sets of unrepeated values:

2 ~ 20+17;   3 ~ 20+10+7 etc.   

The challenge was to make the longest list of values. eight was the longest list: 1, 2, 3, 4, 5, 6, 7, 9 or 2, 3, 4, 5, 6, 7, 8, 9. We thought these were the only ones possible.

Thank you for sharing your thoughts.  I'm not sure I agree that this challenge is impossible to prove, though!  Let's see whether others have any further thoughts ...

Hello, from Mr. Fearn who teachers at East Park Primary School who sent in at a much later date the following:-

I'm a regular user of your fabulous range of problems with my Y5 and 6 classes. This morning, we had a go at 'Make 37' and had a great time with the investigation.
The children quickly realised that adding 10 odd numbers would only ever generate an even total, however, they continued with their strategies and came up with the following suggestion:

3³ + 1⁵ + 5 + 1⁷ + 1 + 1 + 1 = 37

10 numbers have been taken from the bags, imaginatively organised and added to find a total of 37. I was very impressed with their ingenuity !

I am also impressed. I wonder if there are more solution of this kind?

The following three submissions are from Michelle L, Michelle D and Cathy who go to Greenacre Public School Australia:

The problem 'Make 37' requires you to use ten of any of the numbers 1, 3, 5, 7 to make a total of 37. Note that the numbers are all odd and 37 is also an odd number, but you need ten of the odd numbers to make another odd number. 

Usually, an odd number added to another odd number equals an even number. Three odd numbers added together equals an odd number. Four odd numbers added together equals an even number. Five odd numbers added together equals an odd number. Six odd numbers added together equals an even number and so on. When you get up to ten numbers, you will find that the solution is an even number. 37 is an odd number, not an even number so therefore the solution to the problem is that it's impossible.   You can make 37 with nine of those numbers or eleven of them because nine or eleven odd numbers added together equals an odd number.

The solution for "Make 37" is that it is impossible to make 37 with the numbers 1, 3, 5 and 7 if you add them ten times. This is simply because when you add odd numbers an even amount of times such as ten then you result will be an even number. 37 is an odd number so you can't get it with those numbers when you add them ten times. However if you were to add the numbers nine or eleven times then you could make 37. What I noticed about the numbers is that other than the number one the other numbers are prime.

Making 37 out of the numbers 1, 3, 5, 7 is possible if you could pull out the numbers out of the bags nine, eleven or an odd number of times. Since the question gives you the instruction to pull it out only ten times it is therefore impossible. This is because ten is an even number and if you added the numbers an even amount of times, the answer will have to be even. 37 is an odd number. If you added these numbers ten times you would get all the even numbers such as 36, 38 and if you added it an odd amount of time you would get an odd  like 37, 39. 

So there are reasons why it is impossible and a way of getting around that once you know about raising numbers to a power. Thank you all for your contributions.