Math Learning Steps

The value of these games

Strategy games are games where the players have choices that produce
better or worse outcomes. These games have a lot to offer children mathematically, even when
there is no explicit numerical content. In particular, they naturally motivate children to do problem
solving.

An introduction to Nim

Now that your child has been playing strategy games for a while, it is time to look into how to get more out of them. Let’s use the game of Nim as an example. This game has some very simple rules: Pick a starting number, say 10, and a person to go first. Players alternate turns choosing to subtract 1 or 2 from the running total. The person who reaches 0 wins. The arithmetic is simple enough, but the strategy is challenging.

Learning from experience

If you imagine you are playing any game against an extremely capable opponent, the game becomes a puzzle. How can I find the move that let’s me have my best chance of winning no matter how well my opponent plays? One strategy is to play the game many times and pay attention to what seems to work and what doesn’t. This approach is a good start and it does provide the opportunity for observation and insights. This can be a slow method of learning, and it may be very hard to find patterns in complicated games – imagine all the possibilities if we picked the starting number of 100 for Nim!

Solving Nim

Can we look ahead a few moves and figure out good moves that way? That is hard to do starting at 10. If we start at 5, it’s pretty easy to do. If we subtract 2, the new number will be 3. No matter whether the other play subtracts 1 or 2 at this point, we will win. So, if we are at the number 5, we know how to win. What happens with other small starting numbers? Can we find a pattern that will tell us which starting numbers will be winners and which will be losers? Can we explain why that pattern is true?

The goal

If we follow this line of attack, we will completely solve how to play the game of Nim. What worked for Nim may or may not work for another game. This is problem solving, and each new problem can bring fresh challenges and a need for new ideas. And that is the fun. Share this attitude of puzzling and being challenged to your child. As they play a strategy game, discuss with them their ideas about what would make for a better or worse move at that moment. The point isn’t so much to find the perfect move as it is to enjoy looking for it and having a discussion about it with people who they can enjoy sharing ideas with.