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Stage: 2 Challenge Level: Challenge Level:1

We had nearly $100$ solutions from across the world sent in to us. From Hymers College Junior School we had solutions sent in from all of these pupils, Lucy, Max, Abdul, Rikki, Charlie, Helena, Imogen,  Louisa, Helena, Eva, Harley,  Rory, Thomas, Matthew, Khushi, Adi, Amaan, Mariana, Andrew,  Sunnie , Sophie, Amy,  Rashi, Eilza, Signe, Claire, Adnan, Amar,  James, Adam, Amelia, Ayoun and  Amy

Elliott from Solihull School in England sent in a good idea linked with Pascals Triangle;

I've coloured in the even numbers in Pascal's Triangle

In this simple calculation I will explain ........ how adding $2$ odd Numbers and an Even number together always adds up to an even number at the end. First of all I will demonstrate the one on the website as you see it is even
More equations
$5+9+4=18$       and  $9+20101=20110+2=20112$


Amrit from  Farm Nursery, Infant, and Junior School
All even numbers can be represented as $2a$, and all odd numbers as $2b + 1$, where a and b are integers. Thus the sum is $2b + 1 + 2c + 1 + 2a = 2(a + b + c + 1)$. Thus the answer is always even.

Isabel and Caner
 from St Theresa's, Finchley in the UK gave this very thorough explanation

Odd+Odd always Equals even because the next number is always even. For example, if you add $7+3$ you are adding $3$ which is $2$(even)+$1$ and $7$ which is $6$(even)+$1$ even + even = even and $1+1=2$(even) Therefore odd=Odd must always = Even

When you add an even number to an even number it always equals an even number this is because if you add $6$ and $2$ the next number from $6$ is $7$(odd) and the next number from $2$ is $3$(odd) We already proved in our explanation above that odd+odd = even, therefore if you add the 1 more from the even number in both parts of the equation, you will make two, which will combine with the even total of the odd numbers to form another even.

An odd number + an odd number + an even number always equals an even number this is because odd is 1 away from even so it's an even number but when you add an even number + an even number the answer is an even number,when you add an odd number + an odd number the answer is even and when you add even number's together your answer has to be even because your adding an even number to an even number.


Shivek from Monkfield Park Primary School sent in this interesting account;

Example 1.    $5+9 = 14$ ; $14+ 22=36.$
Example 2.    $1+1 = 2.$;  $2+4=6$

Example 3.    $99+99=198$; $198+198=396$.

How  I  did  it? Even numbers end in $0,2,4,6,8$ and odd  numbers end in $1,3,5,7,9$.

Lets add $2$ odd numbers like: $1+3$. You will get $4$
Add $3+5$. You will get $8$  and Add $5+7$. You will get $12$

So, you see: When you add any odd number to another odd number you get an Even number.

Now when you add $2$ even  numbers like $2+2, 4+2 or, 6+4 or 8+4$ you will always get an even number answer. Like ($2+2=4, 4+2=6 or, 6+4=10 or 8+4=12$.)

As you can see above all addition of even numbers ends in $0, 2, 4, 6 or 8$ which are even numbers.

So, Odd Number + Odd Number = Even Number and Even number + even number = Another even number. QED (quite easily done) by Shivek.


Finally from Joanne I think she was rather young and this was written by a helper (?), it says;

Jo used a selection of coloured elastic bands to test out what happens:
$4$ Green bands, $5$ Blue bands, $3$ Yellow bands
GGBBBY  total = $12$ = even

$6$ Green bands, $7$ Blue bands, $7$ Yellow bands
GGGBBBYYYY   total = $20$ = even

She said: 'The number of bands is always even. I put them in $2$ rows and the two rows are the same length, so it is an even number.'

Jo tried a bigger number in her head to test her theory: $10 + 21 + 35 = 66$ = even

Jo tried to work out why her number would be even by working out a rule:
even + odd + odd = even; (even + odd = odd, then odd + odd = even)

She made up these rules for adding two numbers together:
even + odd = odd;  Even + even = even;  odd + odd = even

Jo had a go at trying out what would happen if she changed the odd and even numbers using the rules she had worked out:
even + even + odd = odd;  even + odd + even + odd = even
odd + odd + odd + even = odd (odd+odd=even, then +odd=odd, then +even=odd)


Well done to these and all the others who sent in so very many solutions, most with very good thoughts and understanding.