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Elf Suits

Age 7 to 11
Challenge Level

Sophie wrote to us to say:

I spent a lot of time working on this solution. You should do 4 x 4 x 4 x 4, this equals 256. Therefore the elves would not run out of combinations until 256 days worth of different suits. Me and my school agreed on this as the answer.

I asked Sophie if she could explain a little bit more about why she did 4x4x4x4 as I thought it would help other people reading the solution. She wrote back to me:

The reason I did 4x4x4x4 is because there are 4 elves and 4 different clothing types (bottoms,t-shirt, jacket and hat). I was looking for the most combinations and started by doing it with colour at school. That evening when I went home I went on the website, I looked at the solution to see what I was working with and I realised that I would have to find a new systematic way of working out the problem . The next day at school I tried the 4x4x4x4 way . I don't know why I just sort of looked at a way of getting a big number and using the numbers I had . When I went up to my teacher, she said that I could be right as her husband did a degree in maths and he got the same answer as me. The rest of the class went with that method and we all agreed that it was right . As no-one else in my school was going to email then I would. All I really did was use what I had and apparently the method I used is called the use of powers as it was 4 to the power of 4.

Thank you Sophie for your messages. Yasmin from Bancroft Prep School sent the following which might help to explain why 4x4x4x4 works:

To solve this problem, I started by using just two colours. 
I began with suits all one colour.  Once I had found all two possibilities, I went on to using three of one colour and one of the other.  There are four possibilities for this. 
After that I did the same thing but with the opposite colour.  I found four possibilities. 
Then, I used two of each colour.  I found six possibilities. 
I ended up with 16 combinations for two colours.

I had to think about how you could make 16 with four twos.  Then I spotted:

Hats - 2 colours
Shoes - 2 colours
Jackets - 2 colours
Trousers - 2 colours

2 x 2 = 4
4 x 2 = 8
8 x 2 = 16

After that I used three colours.  Instead of working it out I decided to find out how many combinations there were:

Hats - 3 colours
Shoes - 3 colours
Jackets - 3 colours
Trousers - 3 colours

3 x 3 = 9
9 x 3 = 27
27 x 3 = 81

So I knew there were 81 combinations using three colours.  After that I went on to four colours.

Hats - 4 colours
Shoes - 4 colours
Jackets - 4 colours
Trousers - 4 colours

4 x 4 = 16    
16 x 4 = 64    
64 x 4 = 256

I knew there were 256 combinations for four colours.

The fun then lasted for 256 days until the elves ran out of combinations.

Well done, Yasmin - very clearly set out.