Investiga tion 36
Games: NIM and the 15 Puzzle
By the end of this investigation, you should be able to give precise and thorough
answers to the questions listed below. You may want to keep these questions in mind
to focus your tho ughts as you complete the investigation.
• How is the game of NIM related to group th eory? What is a g ood strategy for
playing N IM?
• What is the 15 Puzz le , and how can the sy mmetric grou ps tell us if a 15 Puzzle is
Games can be fun to play—and, as it turns out, to study. In fact, many ga mes involve ma the-
matical ide a s or can be analyzed using mathematics. In this investigation, we will learn how group
theory c a n be used to determine winning strategies in the game of NIM and to determine if a 15
Puzzle is solvable.
Preview Activity 36.1. Go to any online version of NIM and play the game a few tim es. Search for
a winning strategy.
The Game of NIM
To play the game of NIM, one begins with a number of sets, or stacks, of objects. We can think
of these stacks as piles of stones, as shown in Table 36.1. (Note that we have displayed the piles
horizontally to save space.) In this example, the ﬁrst pile has 6 stones, the second has 2, and th e
third has 3.
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A NIM game.