Three Ways to Play Nim
We wanted to share our 3 favorite ways to play Nim, so we’ve made an interactive Google Jamboard containing all three games. Nim games come from the mathematical field of Game Theory, but they also happen to be a really fun way to explore math with kids!
To make your own copy just click below and then click on the three dots next to the share button and select “make a copy.”
Each of these can be played using household objects, and a hundreds chart.
The underlying goal of playing each game is to figure out a winning strategy, that will work every time. Here are some guiding questions that will help as you search for a winning strategy.
Guiding Questions
Is it better to go first or second?
At what point can you be sure of who will win?
Take 1 or Take 2
This is one of the simplest examples of a Nim game. It’s a 2-player game, where players take turns taking 1 or 2 objects. The last person to take an object loses.
We like to imagine playing with a plate of cookies, and you don’t want to be the one to take the last cookie. Maybe it’s because you think it’s rude or maybe you could say, “whoever takes the last cookie has to make the next batch.”
Once you’ve solved this version of Nim, it's easy to make variations of this by adding or subtracting from the starting number of cookies. In fact, that’s a great way to build up to solving this version, try playing take 1 or take 2 and just starting with 1 cookie, 2 cookies, 3 cookies, etc.
Another way to make a variation is to change the rules of how many cookies you are allowed to take each turn. For example, try playing Take 1, 2, or 3.
Race to 100
Although “Race to 100” is a type of Nim game, by switching up the rules and the format, kids will have to figure out how to apply their reasoning from the previous game to find a winning strategy in this game. Also, by using a Hundreds Chart, there are lots of opportunities for students to look for and make use of mathematical structures. On the Jamboard, we use a transparent circle as a pawn.
Again, it’s easy to make a variation of this by changing the goal number to 20 or 50, and changing the number of steps you can take each turn.
Nim-Piles (2-3-5)
In Nim-Piles we add another layer, which makes it a fair bit more challenging to find the winning strategy. In this version, there are three piles: a pile of 2 cookies, one with 3, and another with 5. Each turn a player must choose a single pile from which to take cookies, but can take as many as they’d like from that pile. Last one to take a cookie loses.
Remember to use the guiding questions to help uncover a winning strategy. Is there some endgame situations where you know who will win? Can you work backwards from there to build a winning strategy that will work from the beginning? Do you want to go first or second to guarantee a win?