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

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Tiles on a Patio

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

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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?

Torn Shapes

Age 7 to 11 Challenge Level:

Another well answered problem! Particularly clear solutions were sent in by James from St Mary's School, Sara-Louise from Perton Middle School and Rowena from Christ Church Infants. Sara-Louise said:

My method for this problem was to count the number of squares along one side, then times them by the number of squares along the other side.
For the rectangles that were made up of two shapes, I first split the single shape into two, timesed the number of squares along the sides of the shapes (separately), then added them together.

Rowena had a slightly different way of finding out the number of squares taken up by each shape. She wrote:

I am going to count the whole squares up and across, then draw the outline of the rectangle, and then draw the squares on to it that weren't there before. I will count the squares and put the total into a table.

Here's Rowena's table:

Rectangle Number of Squares
Orange 3 down x 5 across = 15
Blue 4 down x 8 across = 32
Green 5 down x 3 across = 15
Yellow 5 down by 6 across = 30
Pale orange 21
Purple 18

For the final shape, James said:

There were $3$ in the column for the last puzzle and I could see a bit of a sixth square so the smallest number was $3 \times 6 = 18$. The longest shape had $8$ in a row so the most it could have is $3 \times 8 = 24$.