8. Points are drawn on the sides of a square, dividing each side into $n$ equal parts (so, in the example shown, $n=4$ ). The points are joined in the manner indicated, to form several small squares (24 in the example, shown shaded) and some triangles.
How many small squares are formed when $n=7$ ?
A 56
B 84
C 140
D 840
E 5040
SOlUTION
B

The main diagonals of the square shown in the question divide the square into four quarters each of which contains $1+2+3$ small squares. So the number of small squares when $n=4$ is $4 \times(1+2+3)=4 \times 6=24$, as the question says.

It can be seen from the diagram on the right that, similarly, when $n=7$ the number of small squares is given by $4 \times(1+2+3+4+5+6)=$ $4 \times 21=84$.