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A man paved a square courtyard and then decided that it was too small. He took up the tiles, bought 100 more and used them to pave another square courtyard. How many tiles did he use altogether?

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Squares in Rectangles

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

A $2$ by $3$ rectangle contains $8$ squares:

six small $1\times 1$ squares and two larger $2 \times 2$ squares
A 2 by 3 rectangle


A $3$ by $4$ rectangle contains $20$ squares:

twelve $1\times1$ squares, six $2 \times 2$ squares and two $3 \times 3$ squares

A 3 by 4 rectangle

Consider rectangles with a height of $2$ units.
Increase their width by $1$ unit at a time.
What effect does this have on the total number of squares?

What about rectangles with a height of $3, 4, 5, \ldots$?

Make a note of the number of squares in rectangles with a height of $2$ units.

Do you notice anything special?

Use your results to decide whether a rectangle with a height of $2$ units can contain exactly $100$ squares?

What about rectangles with a height of $3, 4, 5, \ldots$?

Draw up a table of results.