### 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?

### 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?

### Let's Investigate Triangles

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

# 3 Blocks Towers

## 3 Blocks Towers

Take three different colour blocks, maybe red, yellow and blue.
Make a tower using one of each colour.
Here's one with red on top, blue in the middle and yellow on the bottom.

Now make another tower with a different colour on top.

How many different towers can you make?

When you are sure you have found them all, try it with four colours.

You may like to print off one of these sheets for recording three-block towers. Here are sheets for recording four-block towers.

### Why do this problem?

There are two parts to this problem. Firstly, making as many different towers as you can think of, and secondly, making sure that all possible ones have been made. The first is easy, the second requires some systematic working and logical thinking.

### Possible approach

You could use the three block problem as a whole class introduction to the four block one, which children could then go off and try on their own.

Start with a whole class activity in which you invite children to come to the front and make a three-block tower that's different from the one made by the person before. Check in each case why theirs is different, encouraging the children to do the explaining. Keep going until you have six different ones, and then ask another child or two to come and make another and different tower. They will of course find it impossible, which gives you a nice way in to ask "Have we got them all? How can we be sure?".

Allow some time for different children to suggest ideas. Encourage any sort of rearranging which helps the pattern to be clear. If no-one suggests it, point out that we can group them in different ways - with red at the top, blue at the top, etc. and that this helps us to see if there are any missing.

Give each pair three different colours of blocks to replicate the activity and convince you that they have found all possible towers.

When they have done that to your, and their own, satisfaction, suggest that they work in groups of four to produce as many four-block towers as possible, arranging them as they work so that they can be pretty sure that they have found them all. This sheet may be useful if you are likely to run out of multilink. Cutting out and rearranging is a powerful way of organising the thinking.

### Key questions

Have we got them all? How can we be sure?
How could we rearrange them to help?

### Possible extension

Working with four blocks is an extension to this problem anyway, but as an added extension you could again ask if the children are sure that they haven't missed any out, and how they know. Look for different ways of arranging and listen for the explanations that go with them.

### Possible support

Children who find this way of working difficult could be encouraged to colour their solutions on the three-block sheet provided, cut each out and rearrange them into a pattern. Look for other similar problems (such as dressing teddy in red green or yellow hats, t-shirts and shorts) which help to reinforce the importance and use of working systematically.