### Homes

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

### Stairs

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

### Train Carriages

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

# Mixed-up Socks

##### Stage: 1 Challenge Level:

Many of you solved this problem for three pairs of socks. Katarina explained:

1. First, choose three colours of socks.
2. Then, reduce the colours to just their initials, eg. Blue - B. It also makes it easier!
3. Next, have a play around with the letters!

B + G = pair
B + O = pair
G + O = pair

You have to make sure that you haven't repeated a pair or a colour.

What good advice, thank you, Katarina. I agree that checking your solution is very important as it's easy to make a mistake in this problem.

Bafiar, Yunus and Alper (from Private IRMAK Primary and Secondary School Maths Club, Istanbul, Turkey) said :

We discussed your Mixed-up Socks question in our Math club hour and we found the answer using coloured pens. We made a combination using three socks. When we made pairs with red and blue socks, the other green and red socks, so the third pair must be green and blue. We found only one way:

Lucy from Bishop Ramsey School, Priya from Loughborough High School and Katarina all then went on to introduce another pair of socks of a different colour. They each foundtwo ways to mix up the socks this time. However, Bafiar, Yunus and Alper found more:

For four pairs of socks we found three ways. For example if the fourth pair is purple:
1. Green-Blue, Green-Purple, Red-Purple, Red-Blue
2. Red-Green, Green-Purple, Purple-Blue, Red-Blue
3. Green-Blue, Red-Purple, Purple-Blue, Red-Green

I wonder if you can explain how we know we have got all the different ways?