2
Try Small Examples!
If there are 1, 2, or 3 only,
player who moves next wins
If there are 4 chips left,
player who moves next
must leave 1, 2 or 3 chips,
and his opponent will win
With 5, 6 or 7 chips left, the player who moves
next can win by leaving 4 chips
0, 4, 8, 12, 16, … are target
positions; if a player moves to
that position, they can win the
game
Therefore, with 21 chips,
Player I can win!
21 chips
What if the last player
to move loses?
If there is 1 chip, the player
who moves next loses
If there are 2,3, or 4 chips left,
the player who moves next can
win by leaving only 1
In this case, 1, 5, 9, 13, … are a win for the
second player
Combinatorial Games
There are two players
There is a finite set of possible positions
The rules of the game specify for both players
and each position which moves to other
positions are legal moves
The players alternate moving
The game ends in a finite number of moves
(no draws!)
Normal Versus Misère
Normal Play Rule: The last player to move wins
Misère Play Rule: The last player to move loses
A Terminal Position is one where
neither player can move anymore
What is Omitted
No random moves
No hidden moves
No draws in a finite number of moves
(This rules out games like poker)
(This rules out games like battleship)
(This rules out tic-tac-toe)