### Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

### Cuisenaire Rods

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

# Simple Train Journeys

##### Age 5 to 11 Challenge Level:

We had a number of solutions come in with answers from Connor, Isobel, Brian, Josh, Stephanie and Ben.

H.C. from SIS School sent us this very good answer:

I started calculating the number of ways through 4, 5 and 6 stations first.
I did it in an orderly manner.You must start with Dorby and then Ender and Dorby again so you will not miss any of the ways. Then you can start with Dorby, Ender and Floorin and so on ...
Here is the way I used from 4 stations to 6 stations:

4 stations:
1.Dorby-Ender-Dorby-Ender
2.Dorby-Ender-Floorin-Ender
3.dorby-Ender-Floorin-Gambolin

5 stations:
1.Dorby-Ender-Dorby-Ender-Dorby
2.Dorby-Ender-Dorby-Ender-Floorin
3.Dorby-Ender-Floorin-ender-Dorby
4.Dorby-Ender-Floorin-Ender-Floorin
5.Dorby-Ender-Floorin-Gambolin-Floorin

6 stations:
1.Dorby-Ender-Dorby-Ender-Dorby-Ender
2.Dorby-Ender-Dorby-Ender-Floorin-Ender
3.Dorby-Ender-Dorby-ender-Floorin-Gambolin
4.Dorby-Ender-Floorin-Ender-Dorby-Ender
5.Dorby-Ender-Floorin-Ender-Floorin-Ender
6.Dorby-Ender-Floorin-Ender-Floorin-Gambolin
7.Dorby-Ender-Floorin-Gambolin-Floorin-Ender
8.Dorby-Ender-Floorin-Gambolin-Floorin-Gambolin

3rd stat - 2 ways
4th stat - 3 ways (2+1)
5th stat - 5 ways (3+2)
6th stat - 8 ways (5+3)
7th stat - 12 ways (8+4)

This is great - you have really gone about it in a logical way.

So, I wonder if anyone can predict how many ways could be found for 8 stations? This is an example of the Fibonacci sequence. You can find out more about this special pattern of numbers in this article .