8. The diagram shows eight small squares. Six of these squares are to be shaded so that the shaded squares form the net of a cube.
In how many different ways can this be done?
A 10
B 8
C 7
D 6
E 4
Solution
D
Commentary
Answering this question as we do to comes down to checking a list of grids obtained from the given diagram by ignoring two of the squares to see in how many cases they form the net of a cube. There are 28 different ways to delete two of the squares in the diagram. Rather a lot to check! So we begin by thinking of a way to cut down the number of grids that we need to check.

When it comes to checking grids, under the conditions of the SMC you will probably have to use your visual imagination to decide whether they could be folded to make cubes. Outside the SMC you could do a practical experiment.

We label the squares as shown in the figure.
We note first that the net of a cube cannot use both the squares $P$ and $Q$ as these must fold to make the same face of any cube.
Each net includes the square $T$, as otherwise it would not be a connected set of squares. However, no net of a cube can use all four of the squares $T, U, V$ and $W$. Therefore each net must leave out one of the squares $U, V, W$.
It follows that there are only six ways in which we might produce the net of the cube by not using a pair of squares. These involve not using one the following pairs of squares: $P$ and $U, P$ and $V, P$ and $W, Q$ and $U, Q$ and $V, Q$ and $W$. The six grids obtained by ignoring each of these pairs of squares in turn are shown in the figure below.
You should be able to see that each of these is the net of a cube. Therefore there are six different ways in which six of the squares can be used to form the net of a cube.

For inVestigation
8.1 Are there any other nets of cubes other than those shown in the figure above?