'Learning Mathematics Through Games Series: 4. from Strategy Games'

From: http://nrich.maths.org/  4th in a series

The first three articles in this series can be found on My Blog (Why Games?), (Types of Games) and (Creating Your Own Games).

There are several educationally useful ways of incorporating games into mathematics lessons.
Games can be used as lesson or topic starters that introduce a concept that will then be dealt with
in other types of activities. Some games can be used to explore mathematical ideas or develop
mathematical skills and processes and therefore be a main component of a lesson. Perhaps the
most common use of games is for practice and consolidation of concepts and skills that have
already been taught. Yet another way to use games is to make them the basis for mathematical
investigations.

Basic Strategy Games

Basic strategy games are particularly suitable as starting points for investigations because players
instinctively to try to discover a winning strategy, and usually the best way to do this is to analyse
the outcomes of series of 'moves'. With a little encouragement from the teacher, a mathematical
investigation is born. A few questions at the appropriate time will open up the task for the children
and lead to some good quality mathematical thinking.

For example, a basic version of the ancient game of NIM can be used to start an investigation.

          NIM
         Make a pile of seven counters. Two players each take turns to remove either one or two
         counters from the pile. The player left with last counter is the loser .

 

Invite the children to play the game several times and they are sure to begin searching for winning
strategies without being prompted. Ask whether it matters whom goes first and encourage them to record moves. Opponents will soon become partners in investigation as they test their theories.

The teacher's role then is to get the children to explain and justify their strategy and so 'teach' or
convince someone else. Now the game has been mastered it will no longer be enjoyable. It is time
for a "What if??" question.

What happens if you start the game with a different number of counters? (A series of key numbers
will emerge, as well as some interesting observations about odds and evens and multiples).

Problem-Solving Skills

Analysing a game in this way will typically engage the student in some highly desirable problem
solving strategies and processes: -

• being systematic,
• transforming information, (e.g. inventing a method for recording moves), 
• searching for patterns,
• applying mathematics (calculations, algebra),
• manipulating variables,
• working backwards, simplifying the problem,
• hypothesising and testing, and
• generalising (perhaps even producing a formula)

What if you can take a different number of counters away?

Variations of the Same Game

A way to take the investigation further and hence the mathematics further is to introduce a
variation of the game and search for winning strategies.

          NIM 3, 4, 5
          Make a row of 3 counters, a row of 4 and a row of 5. Two players each take turns to remove
          any number of counters from a particular row. The player left with the last counter is the      
          loser (or winner, as agreed at the start).

It might be helpful to suggest simplifying this game to a configuration of seven counters.

This may help them realise the importance of groups of two of four in the analysis.

It is often the case that problems and puzzles that appear to be quite different have a very similar
underlying mathematical structure. This can also be the case for strategy games.

          SLIDE (Linear NIM)

          Place a counter on each of the four coloured squares. Two players take turns to move any
          counter one, two or three spaces, until they reach the end of the track and are removed. No
          jumping is allowed. The winner (or loser as agreed) is the person left with sliding the last
         counter of the track. 

To analyse the game, it is helpful to start by playing it with only one counter, then two, then three.
Clear strategies can be found with one counter, but the introduction of other counters allows
blocking, which complicates the moves.

How can what has been discovered about these games be used t o create new challenging games?

The NIM game can also be extended into a two-dimensional game board.

          MINIM (2-D NIM)

          Place twenty-five counters on the game board as shown. Players take turns to remove one   

          or more counters that are side-by-side (no spaces between) on a straight line. The last player
          to take a counter is the loser .

Though complete analysis is too difficult, continuous scoring will help focus attention on early
moves. (1 point for each counter removed, minus 5 for the last counter). Encourage the children to
think backwards form the final move to discover helpful strategies towards the end of the game.

More Games?

All the games published monthly on the NRICH site are accompanied by questions that prompt
mathematical thinking and investigations.

An excellent source for groups of related strategy games is a book called "Strategy Games" by R.
Sheppard and J. Wlikinson. It is published by Tarquin and available through their website.