### Number Detective

Follow the clues to find the mystery number.

### Score

There are exactly 3 ways to add 4 odd numbers to get 10. Find all the ways of adding 8 odd numbers to get 20. To be sure of getting all the solutions you will need to be systematic. What about a total of 15 with 6 odd numbers?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

# Make 37

##### Stage: 2 and 3 Challenge Level:
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 ...

Swift Class at Rudston Primary sent in the following:

Make 37 really got us thinking. We worked with a partner and began adding using a running total and then some of us were grouping 1's, 3's, 5's, and 7's together and multiplying.

We showed great resilience and kept on trying out ideas. Two groups even spent their playtime looking for a solution.

Groups who thought they had succeeded always had 9 or 11 numbers, not 10. This made us ask, is it possible to make an odd number with an even number of odd numbers?  We believe this problem is impossible and that's why there are no solutions on the solutions page!

From Josephine at St Wilfrid's Ripon we had:

This task is impossible because if you add an odd number to an odd number you will always get an even number. We would need an odd number and an even number to make an odd number.

eg: 7+7=14, even     4+5=9, odd
9+9=18, even     3+2=5, odd
7+3=10, even     6+7=13, odd

If you use nine numbers it would be possible because nine is an odd number so you will have an odd number of odd numbers. eg:1+1+1++7+5+5+5+5+7=37, nine numbers. You wouldn`t be able to do it if you had an even number (e.g. eight) of odd numbers because that would make four pairs of odd numbers, each making even numbers and thirty seven is an odd number.

Arshiya from Monkfield Park Primary School wrote;

The sum asks us to make an odd number total using odd numbers. We know that if you add two odd numbers they will make an even number. The pattern is such that if you add two odd numbers the answer is even, three odd numbers gives odd number as total and so on.

For example:

1+3=4
1+3+5=9 (odd number of terms)
1+3+5+7=16 (even number of terms)
1+3+5+7+9=25

Therefore taking ten odd numbers from the bag will only give a total number which is even and not 37 or any other odd number.

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 number like 37, 39.

Julian Yu from British School Manila Philippines wrote:

It is impossible to pick ten odd numbers such that their total is an odd number.

Proof:
All odd numbers can be expressed as 2k+1 for some integer k.
Let the ten odd numbers be 2a+1, 2b+1, 2c+1 ... 2j+1.
Their sum is 2(a+b+c+...+j)+10
Which is equal to 2(a+b+c+...+j+5)
Which is always even.

Thank you too to Haider, Isobel, Matthew, Kanika and Arjan all at St Matthew's C of E Primary School, Redhill, Surrey and Edward, Callum, Eleanor, Elliot, Henry, Jamie, Neave and Rebecca from Burford School.

Thank you for those excellent thoughts and things that you noticed.