### 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?

# Walking the Squares

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

Haaniah from Woodthorpe JI School sent in this good solution;

I first thought about how to get to the middle using any numbers of squares, for example, clockwise. Then I decided to go clockwise, using two squares and two rows at a time. When I got to the corner I decided to enter the corner diagonally. When I got to the final three rows, I decided to still go clockwise using two squares at a time, except this time I used three rows at a time.

First, I imagined how you could use the most number of tiles up using more than two of the same colour; you would follow it all the way around, clockwise. Then I decided to try two rows at a time for example: You start at the top right orange corner. Go right 1 then down 1, right 1 and up 1.

Continue until you reach the corner. Enter the corner diagonally then 1 across and exit diagonally as demonstrated in the picture when you have gone the whole way round enter the next rows diagonally and repeat, this time three rows at a time. When there is only three squares left, enter straight into the middle.

Daya from Brackenbury Primary School sent in these as their progression:

Megan from  St. Oswalds Primary School sent in her solution:

Swasti  Year 5 from Maybury Primary School, Woking sent in the following: with following as a start of her excellent explanation, the full text can be viewed here.docx

So, firstly, I worked out how many tiles the square has (121) as it would be the maximum number of tiles to walk on.
Then, I used a zig-zag pattern (as shown below) to walk on the square until I met a corner.
When I met a corner, I just realised that in order to walk on the maximum number of tiles, I couldn’t use the zig-zag pattern as I did earlier because then I would have to walk on a same coloured tile more than twice.

Thank you for all these solutions that you sent in, if any other pupils can find a way of leaving even fewer empty squares then please send them in.