'A great problem that instantly engages and gives rise to lots of algebraic sequences - terrific for linking sequences to real situations'
Imagine a 3D game of Noughts and Crosses, Tic Tac Toe or OXO for short! How many ways are there to get three in a row? How can you count them systematically, making sure you don't count any twice or leave any out? What about a bigger playing cube? What about a bigger playing cuboid? What impact does the length of the winning line have? These questions are the tip of the iceberg.
This classic problem can begin in 2 or 3 dimensions in many different ways and lends itself very well to some algebraic sequence modelling. Imagine being able to predict the number of winning lines as the problem gets more and more complicated. The model in the picture was carefully constructed with stuff lying about the classroom in just 10 minutes to prove that it can be done! Play the game and have a go!
This activity can run from the description above at its most open ended or broken down into different starting points using the OXO worksheet. OXO Teachers notes offers some helpful thoughts about how this activity might best work in the classroom. See below for some photos and video that help run this activity.
The following are in no particular order and could all be starting points for investigation.
Try the 3D cube case. The following video shows a model that can be built quickly to get the idea going. So imagine a 3 x 3 x 3 playing cube with 3 in a row needed to win. How many ways are there of winning? Describe how they can be counted. Now think about a 4 x 4 x 4 grid and so on. Think about generalising!
The following is a link to a nice online simulation of this 3D game that can be played by 1 or 2 players. Below is a screen cast of the game in action.
'2D' - Square grids
Watch the video below for an introduction to the simple OXO starting point. How many ways to win an ordinary game are there? What if the grid was 4 x 4 and you needed 4 in a row to win? Can this case be generalised?
'2D' - 3 in a row
Still in 2 dimensions and still squares, but here the length of the winning line stays as 3. What happens now?
'2D - Recatangles!'
Now unleash the rectangles! Keep the length of the winning line at 3 to start with then explore different length winning lines!
Here follows an outline of what the task is. There are various different starting points for this activity. Teachers can find advice on this in the teachers notes.
- Define a starting point. This means a game of noughts and crosses. Describe the defining features of the game and ask how many winning lines are possible.
- Pick a variable from this starting point that can be changed. Change it a few times and note what it does to the outcome.
- Make a conjecture about the way the number of winning lines changes as the variable is changed
- Attempt to generalise about this case
- Consider changing a different variable and/or defining a new game and repeat the exercise
- Work towards the most versatile generalisation possible and take care to explain thoroughly the thinking at each stage