### 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?

# Fault-free Rectangles

##### Age 7 to 11 Challenge Level:

Sammy and George from Ardingly, and Bilge, age 12 all sent in correct answers to the first part of the question.

As George explains:

A rectangle doesn't necessarily have to be a oblong it can be a square. All you have to do is build a fault free square which is easy:
You have to put the white square in the middle and then place the red oblongs (so that there are no faults) around the edge of it.

The fault-free rectangle looks like this:

Congratulations to V. Abhijit, age 15 who sent in a fuller solution:

The key behind making a fault free rectangle is this:
Start with a 1 by 1 rectangle, place it at the centre
Now, place 4 2 by 1 rectangles around the 1 by 1 rectangle This will form a fault free rectangle(actually a square, but nevertheless its a rectangle!!)
Thus if you start with a 2 by 1 rectangle, place 2 of them at the centre and then place 4 3 by 1 rectangles around the initial arrangement.

This rectangle is the one made by the red and green rods:

V. Abhijit goes on to generalise...

I noticed that you get a fault free rectangle when I used an "N by 1" rectangle along with an "N+1 by 1" rectangle. I had to put N "N by 1" rectangles at the centre (that forms a square) and then had to place 4 "N+1 by 1" rectangles around the arrangement to get a fault free rectangle(that turned out to be a square)

V. Abhijit gives the smallest fault-free rectangle that can be made using 2 by 1 rectangles as that shown below:

Can you find another?