Nim Games

I enjoy going to schools to give talks. Generally, I try to focus these talks around mathematics that’s not generally taught in classrooms to try to connect to some of the inquisitive nature of the students. One of my favorite ways of doing this is through combinatorial games. These combinatorial games are generally two player sequential games (i.e. players alternate taking moves) where both players know all the information about the game before any moves are made. This is called a game of complete information. In addition, these games are deterministic, in that unlike a game of poker or dice there is no random element introduced into the game.

One of the most common ways of introducing students to combinatorial games is through the game of Nim (which is also called the Subtraction game). I’ve written a script here to help introduce this game. In the game of Nim, there are initially a number (p) of rocks in a pile. There is also an array of possible legal moves that each player can choose from on each turn. Players alternate removing a legal amount of stones from the pile until some player is unable to make a move, at which point the opposing player (the player who made the last move) is declared the winner.

So example a game of (1-2-3)-Nim could go as follows. Suppose initially there are 23 stones.

Stones Player Removed
23 1 3
20 2 1
19 1 2
17 2 2
15 1 1
14 2 3
11 1 2
9 2 1
8 1 3
5 2 2
3 1 1
0 2 1

In the above example, since player 2 removes the last stone, player 1 is unable to move so player 2 is declared the winner. Each move that a player makes is either removing 1, 2 or 3 stones as we initially stated in the rules of the game.

Because Nim is a game of perfect information, we know a lot about the game before any moves are made. In fact, we can determine who should win the game if it is played perfectly just by knowing the set of available moves and the number of stones in the pile. We can do this by considering a game with 0 stones and determining who would win this game (player 2), and increasing the number of stones in the pile one by one and at each new cell, determining who would be the winner. In this method, we can say that we are in a winning position if there is a feasible move that would put the opposing player into a losing position. Consider the following table for the (1-2-3)-Nim game:

Stones 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Winner 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1

We can analyze this table as follows. With 0 stones, there are no moves that any player can make, but since player 1 goes first, they cannot make a move and lose the game. When there is 1, 2, or 3 stones, then player 1 can remove all the stones in the pile and in all cases player 2 will be looking at a situation where there are no stones to remove. When there are 4 stones, no matter how many stones player 1 removes, player 2 will be able to remove the remaining stones to ensure that player 1 is looking at a situation with no stones. We can repeat this process with any number of stones and we arrive at a table similar to the one listed above.

I have a script at my Nim games page where the set of possible moves and the number of stones in the pile are generated randomly and users get to play against a computer. Check it out and let me know what you think.