9. The diagram shows an $n \times(n+1)$ rectangle tiled with $k \times(k+1)$ rectangles, where $n$ and $k$ are integers and $k$ takes each value from 1 to 8 inclusive.
What is the value of $n$ ?
A 16
B 15
C 14
D 13
E 12
SOlUTion
B

The total area of the rectangles of size $k \times(k+1)$, for $k=1,2,3,4,5,6,7,8$, is
$$
\begin{aligned}
1 \times 2+2 \times 3+3 \times 4+4 \times 5+5 \times 6+6 \times 7+7 \times 8+8 \times 9 & =2+6+12+20+30+42+56+72 \\
& =240 \\
& =15 \times 16 .
\end{aligned}
$$

Therefore $n=15$.
In the context of the SMC the above calculation is sufficient to show that, if the smaller rectangles tile a rectangle of size $n \times(n+1)$, for some integer $n$, then $n=15$.
However, for a complete solution it is necessary to show that the eight smaller rectangles can be used to a tile a $15 \times 16$ rectangle.
It looks from the figure in the question that this is possible. The figure on the right confirms that the sizes of the rectangles are correct.
Note also that from this figure we can see directly that the large rectangle has size $15 \times 16$.