### 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?

# Four-triangle Arrangements

## Four-triangle Arrangements

Well, equilateral triangles are great fun to play around with (try Triangle Animals if you haven't already!) but let's not forget the right-angled triangle - particularly the kind that comes from cutting a square in half through a diagonal.
We could take 4 of these and have something like this:

So we can make some rules about how we can re-arrange these four triangles.
Here's a usual rule - EACH SIDE MUST MATCH UP TO A SIDE THAT'S JUST THE SAME LENGTH AND THEY MUST HAVE THEIR VERTICES TOUCHING.

The four arrangments above would obey the rule. But the next two would NOT obey the rule. Can you say a reason why?

So, using plastic, paper, card or other triangles, what arrangements can you make with four right-angled isosceles triangles like the ones at the start?
You will have to decide about allowing "flipping over" or not.

Like in so many investigations it's good after a while to change a bit of the rule and start again.
So let's say that the four must be joined together BUT you can have them joining with one vertex and all OR part of a side touching.

For example the red and orange ones we've already seen above are now allowed:

Others might be: