Intent

In this activity, students explore the idea of a winning strategy in another context.

Mathematics

The game of Linear Nim is distinct among other unit activities in that it does not involve chance. Variations of Nim have been traced far back in history. The game closely resembles the ancient Chinese game of Tsyanshidzi, or "picking stones,” though its origin is uncertain. The earliest European references to Nim are from the beginning of the 16th century. Charles L. Bouton of Harvard University, who also developed a complete theory of the game in 1901, coined the name Nim. He never fully explained the origins of the name, which can be traced to German roots (nimm! meaning "take!") or the obsolete English verb nim of the same meaning. Notice that the word NIM upside down reads WIN.

Progression

Introduce students to the game approximately one week before their analyses and write-ups are due.

Approximate Time

15 minutes to introduce the game

1 to 3 hours for activity (at home)

20 minutes for presentations

Classroom Organization

Pairs to initiate the activity, concluding with presentations and class discussion

Doing the Activity

Allow some time for students to play the game in pairs.

During the week that students work on this activity, you may want to carve out a few minutes here and there for them to play the game again and to talk about their strategies. Ask, When do you know you have won the game?

The day prior to scheduled presentations, identify volunteers to present their findings. Give them transparencies and pens (or other resources) to prepare for their presentations. Some students may prefer to create a computer-based slideshow.

Discussing and Debriefing the Activity

Begin the discussion by having two or three students present their findings. During the follow-up discussion, focus on the theoretical analysis of specific strategies. It’s reasonable to expect most students to develop a complete strategy for the original game, and some may even develop a more general analysis.

One way to conclude the discussion is with a brief Nim tournament, randomly varying the initial and maximum values during the tournament. You will know that students really understand the game when they can determine as soon as the game is set up which player—the first or the second—will win. Their sense of accomplishment can be enhanced by using a random number generator to choose the parameters for each new game.