### 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?

# Three Ball Line Up

##### Stage: 1 Challenge Level:

We received many solutions to this problem, but Charles from Clear Water Bay School in Hong Kong, Alex from Longsands School and Maria from Wimbledon High Junior School took the trouble to explain how they worked it out. Charles says:

If the green ball is in the first place, there are two combinations i.e. red ball second, blue ball third or blue ball second, red ball third.
That gives us two ways if the green ball lands in the first place.
It is the same with the other two balls. Therefore, $3 \times 2 = 6$ ways.

Here are the ways that all three of them listed:

Green, red, blue
Green, blue, red
Blue, red, green
Blue, green, red
Red, blue, green
Red, green, blue

There is only six because there are three of each colour, and if the first colour is red, then the second will be green or blue (and so on) so there are only three starting colours but two of each one.

First I put the blue ball first then I realised that I could only put the red or green next.
Then I did it with the other two colours:
So the answer is there are 6 different ways the balls could land.

Henry at Applecroft School said:

This is how I worked it out:
First, I found a ball, for example green, then I placed it in position 1.
Then I placed the other balls in positions two and three.
Keeping the same ball in position one, then I changed the red and blue balls to make the second combination.
I then placed a different colour ball in position one and changed the balls in positions two and three again.
I completed the task by placing the third colour in position one and changed the balls in position two and three again.

Aiste from  VDU "Rasos" gymnasium, Lithuania wrote:

It's really simple. I can offer the best way to solve this problem. There are three balls and also three spaces for them.
So: _ _ _
We should put three diferent balls.
In the first place you can put all three balls. In the second place - 2 and at last in the third - 1.
So when we multiply these numbers and then will find the answer. $3{\times}2{\times}1$ is 6.

I wonder why we multiply the numbers?

Well done all of those that sent in the solution. There were responses from the United Kingdom, Lithuania, Singapore, Hong Kong and Prague. It's so interesting to know that in many corners of the world pupils are looking at the same challenges and working on them in similar ways. Keep them coming in and if you are viewing  this from a country that's not mentioned then why not submit your solution?