# Ribbon Squares

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

Imagine a square swimming pool with 24 single tiles around it, like the one in the diagram.

Two children stand on different tiles, and hold a ribbon or ribbons across the pool.

Each child can hold one or two ribbons at a time.

Each ribbon runs from the middle of the tile that the child is standing on.

The children are trying to make squares with their ribbons that they call 'ribbon squares'.

Here's a ribbon square they made. It has an area of 9 square tiles.

What are the smallest and largest ribbon squares they can make?

How many differently sized ribbon squares can they make?

What happens if the square swimming pool is made from 20 tiles?

How many ribbon squares can they make then?

Two children stand on different tiles, and hold a ribbon or ribbons across the pool.

Each child can hold one or two ribbons at a time.

Image

Each ribbon runs from the middle of the tile that the child is standing on.

The children are trying to make squares with their ribbons that they call 'ribbon squares'.

Here's a ribbon square they made. It has an area of 9 square tiles.

Image

How many differently sized ribbon squares can they make?

What happens if the square swimming pool is made from 20 tiles?

How many ribbon squares can they make then?

*This problem featured in a preliminary round of the Young Mathematicians' Award.*

An excellent solution came from Sumair and Ayush who are at the American Embassy School, New Delhi in India. I strongly recommend you watch their excellent associated video (see link below).

We started by thinking about the 24 tile pool. The biggest ribbon square we could make was 26 square tiles, and the smallest was 1 square tile. We made 9 different sized squares: 1, 2, 4, 8, 9, 16, 18, 20 and 26.

In the 20 tile pool, the biggest ribbon square we could make was 17 square tiles, and the smallest was 1 square tile again. With this pool, we made 7 different sized squares: 1, 2, 4, 8, 9, 13 and 17.

Please watch our video to find out how we solved this question. We used three different strategies to make squares. Here is the link to our video:

https://www.youtube.com/watch?v=aMYEyqlCgmM

A weird thing we noticed was that the biggest ribbon squares in both pools were 1 sq. tile more than the area of the pool! And we really don't know why that happens, and we want to find out! Another thing we want to find out is more about how the Pythagorean Theorem works.

Millie and Kayne; Bethany and Francesca; Ellie and Alyssa; Lucas and Toby; Bethany and Francesca all from Bradley Green in UK sent in a variety of solutions. Thank you, well done.

Bertie, Mateusz, and Hannah from St. Joseph's Portishead UK also sent in some lengthy explanations. Well done.

Jack from the Tanglin Trust School in Singapore wrote:-

I started off by trying to make the children stand as close to each other as possible but still make squares. I did this by drawing the pool out on a white board.

I found that the smallest ribbon square that is possible is one that has an area of one tile as if the children each stand on a tile where there is another child on the one next to them all the ribbons will be really close to each other and I will be a one tile by one tile square.

Next I tried to to make the children as far apart as possibly to make the largest one possible.

I found that the biggest one possible would have an area of twenty tiles as if each child stood in a corner of the pool and held two ribbons, one in each direction making a right angles he square would be five tiles by five tiles. If they aren't allowed to stand in the corners however, the largest possible square would have an area of sixteen tiles as there would be two children on each side of the pool and they would have three whole tiles in between them so the square would be four tiles by four tiles. Thank you for reading.

Well done all of you for your submissions. If you are looking at these and you have not done the challenge then have a good look at the task. Once you've explored that, you could ask yourself "I wonder what would happen if we changed one of the rules?" and explore further!

### Why do this problem?

This problem gives pupils a challenge in both spatial and numerical problem solving. It is also quite a challenge for recording methods. It is probably best suited for a group of children who are confident problem solvers.The activity also opens out the possibility of pupils asking “I wonder what would happen if . . .?” Pupils' curiosity may be easily aroused while finding solutions to the challenges.

### Possible approach

It would be good to have a square grid background and then have coloured squares arranged for the tile surround.

This could be on card on the floor for the children to see, or represented on an interactive whiteboard.

Some time needs to be spent arranging ribbons/string/lines that obey the rules and produce a square so that learners can internalise the rules.

Give children time to explore for themselves and intersperse this with some 'mini plenaries' to exchange ideas.

It may be helpful for learners to have a printed copy of the task.

This could be on card on the floor for the children to see, or represented on an interactive whiteboard.

Some time needs to be spent arranging ribbons/string/lines that obey the rules and produce a square so that learners can internalise the rules.

Give children time to explore for themselves and intersperse this with some 'mini plenaries' to exchange ideas.

It may be helpful for learners to have a printed copy of the task.

### Key questions

How are you working out the area to show they are not the same?

How have you decided that this is the biggest/smallest?

How have you decided that this is the biggest/smallest?