Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface of the water make around the cube?
It is known that the area of the largest equilateral triangular
section of a cube is 140sq cm. What is the side length of the cube?
The distances between the centres of two adjacent faces of another
cube is 8cms. What is the side length of this cube? Another cube
has an edge length of 12cm. At each vertex a tetrahedron with three
mutually perpendicular edges of length 4cm is sliced away. What is
the surface area and volume of the remaining solid?
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
As well as giving students an opportunity to visualise 3-D solids, this problem provokes the need for students to work systematically. Counting the winning lines in an ad hoc way will result in double-counting or missed lines, with students getting many different answers. It is only by working in a systematic way that students can convince themselves that their answer is correct. By offering a
variety of methods, we hope students will evaluate the merits of the different approaches, and recognise the power of methods which make it possible to generalise.
"If I played a game of noughts and crosses, there are eight different ways I could make a winning line. I wonder how many different ways I could make a winning line in a game of three-dimensional noughts and crosses?"
The image from the problem could be used to show one example of a winning line.
Give students time to discuss with their partners and work out their answers. While they are working, circulate and observe the different approaches that students are using, and challenge them to explain any dubious reasoning. After a while, stop the group to share their results, perhaps writing up all their answers on the board (it is likely that there will be disagreement!).