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Why do this problem?
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!).
"It's often difficult to know we have the right answer to a problem like this, because there is a danger of missing some lines or counting some lines twice. Here
are the systematic methods that four people used to work out the number of winning lines. For each method, try to make sense of it, and then adapt it to work out the number of
winning lines of 4 marbles in a 4 by 4 by 4 cube."
The methods are arranged two to a sheet, so you could give each half of the class a different pair of methods to work on, or alternatively you could give everyone all four methods.
"Once you have adapted the methods for a 4 by 4 by 4 cube, have a go at working out what would happen with some larger cubes, and perhaps try to write down algebraically how many lines of n marbles there would be for an n by n by n cube."
Bring the class together and invite students to present their thinking, by asking them to explain how to work out the number of winning lines in a 10 by 10 by 10 version of the game.
Finally, work together on creating formulas using each method for the number of winning lines in an n by n by n game (or gather together on the board the algebraic expressions they found earlier) and verify that they are equivalent.
How can you categorise the types of winning line, to make sure you don't miss any?
How would you extend Caroline's (or Grae's or Alison's or James') method to count the number of winning lines in a 4 by 4 by 4 cube?
Extend the cubic 'grid' to a cuboid, possibly 4 by 3 by 3 to start with, and ultimately $n$ by $m$ by $p$, always looking for lines of 3 - unless students want to look for other length lines (they could look for lines of 2 on the 3 by 3 by 3 grid).
offers students the opportunity to work with the structure of a cube and consider faces, edges and vertices.