11. Rahid has a large number of cubic building blocks. Each block has sides of length $4 \mathrm{~cm}$, $6 \mathrm{~cm}$ or $10 \mathrm{~cm}$. Rahid makes little towers built from three blocks stacked on top of each other.
How many different heights of tower can he make?
A 6
B 8
C 9
D 12
E 27
Solution
C
ComMENTARY
The most straightforward approach here is to list all the possible ways of choosing three of the blocks. For a full solution it is important to do this in a systematic way that makes it clear that every possible case occurs in the list, and no case is listed more than once. We have set out the table in the solution below in a way that we hope makes it clear that we have achieved this.

In the table below we list all combinations of three blocks each of side length $4 \mathrm{~cm}, 6 \mathrm{~cm}$ or $10 \mathrm{~cm}$. In the last column we have give the height of the tower that the given blocks make.
\begin{tabular}{cccc}
\hline \begin{tabular}{c} 
number of \\
$4 \mathrm{~cm}$ blocks
\end{tabular} & \begin{tabular}{c} 
number of \\
$6 \mathrm{~cm}$ blocks
\end{tabular} & \begin{tabular}{c} 
number of \\
$10 \mathrm{~cm}$ blocks
\end{tabular} & \begin{tabular}{c} 
height of \\
tower
\end{tabular} \\
\hline 3 & 0 & 0 & $12 \mathrm{~cm}$ \\
0 & 3 & 0 & $18 \mathrm{~cm}$ \\
0 & 0 & 3 & $30 \mathrm{~cm}$ \\
& & & \\
2 & 1 & 0 & $14 \mathrm{~cm}$ \\
2 & 0 & 1 & $18 \mathrm{~cm}$ \\
1 & 2 & 0 & $16 \mathrm{~cm}$ \\
0 & 2 & 1 & $22 \mathrm{~cm}$ \\
1 & 0 & 2 & $24 \mathrm{~cm}$ \\
0 & 1 & 2 & $26 \mathrm{~cm}$ \\
1 & 1 & 1 & $20 \mathrm{~cm}$ \\
\hline
\end{tabular}

We see that there are ten ways in which Rahid can choose three blocks with which to make a little tower. However, two of them make towers of the same height, namely $18 \mathrm{~cm}$. So there are 9 different heights of tower that Rahid can make.
For INVESTIGATION
11.1 How many different heights of tower can be built using four of the blocks?