Elf suits
Problem
Mrs Claus made four of the workshop elves new suits for Christmas.
Each suit had matching trousers, waistcoat, jacket and hat.
One suit was all red, one was all green, one was all yellow and one was all blue. The elves were delighted with their presents, but decided it would be fun to wear a different outfit every day for as many days as possible. So they agreed to swap around parts of their suits until they ran out of new combinations.
How many days did their fun last?
At the end of 2015 Lucy and Reece from Eskdale Academy solved the problem and then asked
The answer (?) takes into account all combinations of clothing but it does not take into account whether or not you put the same colours on the elves on the same day. We thought that it wouldn't work because there is only one item of each colour, e.g. one pair of red trousers. Also, is there a system to find out exactly what the elves wore each day?
So, take the challenge further and make some suggestions to Lucy and Reece.
Getting Started
How do you know you've found them all?
What outfits can there be all with blue trousers ... green trousers ...?
Student Solutions
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.