### 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

##### Age 5 to 7 Challenge Level:
These solutions (from a range of countries!') demonstrate some good thinking.

William from Beverly Farms Elementary in the U.S.A. wrote:

First I put red on top and did red white green and then I did red green white.
Next, I did green red white and then I did green white red.
Finally, I did white red green and then white green red.

Manisha from the Dehradun Public School in India sent in the following:

In 3 Block Towers puzzle, I first kept any one colour at the bottom and remaining at the 2nd and 3rd position that made one tower then I changed the 2nd and 3rd block means 2nd at 3rd position (top) and 3rd at the 2nd position that became the 2nd tower.
In this way, now I have put other colour block at the bottom that was the 3rd tower and changed the position of the 2nd and 3rd block so got the 4th tower.
In this way, now I have put the remaining colour block at the bottom that was the 5th tower and changed the position of the 2nd and 3rd block so got the 6th tower.
In this way, we can make a total of 6 towers.

Phoebe and Cai from Ysgol Porth y Felin in Wales had their work sent in:

Phoebe chose to work out her problem by colouring the squares in her book with the options.
On the second part of the challenge, Cai realised he didn’t have to draw each option in his book, once he figured out green on top had 6 different options, he used his times tables to work out the possibilities.

(Click on these images and a larger version will open in a new page.)

Lucy from Hawkesbury Primary  wrote:

Lucy solved this by trial and error, quickly spotting that after finding three possible options, that red had been in each place once, but the other colours hadn't yet been in each place, so there must be more solutions.

Sergio from El Centro Ingles in Spain  sent in the following:

I did first with red at the top and this is how you can do it: (red, yellow and blue) and (red, blue, yellow).
When we finish with red at the top you can do yellow or blue. I am going to do the yellow and these are the orders: (yellow.red.blue) and (yellow, blue, red)
And for finish we have the blue at the top: (blue, yellow, red) and (blue.red.yellow)
That is how you do this.