Marbles in a Box

Problem | Teachers' Notes | Solution | Printable page |
Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem is a 3D version of noughts and crosses. It is useful for encouraging students to work with / visualise 3D solids.


Possible approach

"If I played a game of noughts and crosses, there are eight different ways I can make a winning line. I wonder how many different ways I can 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. It is likely that there will be disagreement, so insist that students explain how they got their answers. It may be necessary to introduce some clear ways of counting without double-counting, in order to reach consensus. One way is to consider where winning lines could begin and end:

"Winning lines can start at a vertex, or on an edge, or on a face.
How many winning lines are there that start (and end) at a vertex?
How many winning lines are there that start (and end) on an edge?
How many winning lines are there that start (and end) on a face?"

Give students some time to think about these three questions, and then discuss their responses.

 "Imagine we played a 4 by 4 by 4 version and needed to get 4 in a row to win.
Or a 5 by 5 by 5 version, and needed 5 in a row to win? I'd like you to work on some different versions of the game until you are confident that you can always work out how many different winning lines there would be. In a while, I'll choose a much larger game size at random, and you'll need to have an efficient method of working out the number of different winning lines."

 

Bring the class together and challenge them to explain how they can work out the number of winning lines in a 10 by 10 by 10 version of the game. Depending on the students' experience of working with algebra, you could work together on creating formulas for the number of winning lines in an n by n by n game.

 

Key questions

Where, exactly, is that line?
Are there any other lines like it?
How many? Where?
Why are there 4 of those lines but 12 of these lines?
How do you know you have got them all?
Have you counted any twice?

Possible extension

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).

Possible support

It is worth developing at least one visual aid. The different layers could be different colours, or referred to as floors in a house. Multi-link may be appropriate, or clear plastic trays from science, stacked 3 high, with the grid drawn on their bases.

With some groups, it might work to make the classroom into the 3D grid, (splitting the space into 27 notional 'cubelets') then letting half the class play against the other half. Ask a couple of pupils to record on the board, where the noughts and crosses were placed, and decide with the group an appropriate 2D representation of the 3D game. It may be an appropriate time for a discussion of the merits and drawbacks of 3D co-ordinates in this context.

Published February 2003.