Squares in Rectangles
A $2$ by $3$ rectangle contains $8$ squares:
six small $1\times 1$ squares and two larger $2 \times 2$
squares
A $3$ by $4$ rectangle contains $20$ squares:
twelve $1\times1$ squares, six $2 \times 2$ squares and two $3
\times 3$ squares
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.