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This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?


Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Ribbon Squares

Age 7 to 11
Challenge Level

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:

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!