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?
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?
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
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.
These printable worksheets may be useful: Marbles in a Box
Marbles in a Box - Methods
"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!).