### Counting Counters

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

### Cuisenaire Squares

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?

# Crossings

##### Age 7 to 11Challenge Level

In this problem we are looking at sets of parallel sticks that cross each other.

There are four red sticks which make four crossings.
How many crossings do the five green sticks make?

Still keeping two sets of parallel sticks, this time with seven sticks in total, can you arrange them in another way, to get a different number of crossings?
What is the least number of crossings you can make?
What is the greatest number of crossings you can make?
Can you find all possible numbers of crossings with seven sticks?
What do you need to do to prove that you have them all or how could you show that you have them all?

Can you find the least/greatest number of crossings for ten sticks?
Can you find the least/greatest number of crossings for fifteen sticks?
Can you predict the least/greatest number of crossings for fifty sticks?
Can you predict the least/greatest number of crossings for any number of sticks?