Introduction to Combinatorial
Game
Acm Honored Class
Weitao
Sprouts
Rules
The game is played by two players, starting with a
few spots drawn on a sheet of paper. Players take
turns, where each turn consists of drawing a line
between two spots (or from a spot to itself) and
adding a new spot somewhere along the line. The
players are constrained by the following rules.
Rules
The line may be straight or curved, but must not touch or
cross itself or any other line.
The new spot cannot be placed on top of one of the
endpoints of the new line. Thus the new spot splits the line
into two shorter lines.
No spot may have more than three lines attached to it. For
the purposes of this rule, a line from the spot to itself counts
as two attached lines and new spots are counted as having
two lines already attached to them.
In so-called normal play, the player who makes the last
move wins. In misère play, the player who makes the last
move loses.
Combinatorial Game
There are two players.
There is a set, usually finite, of possible positions of the game.
The rules of the game specify for both players and each position which moves to
other positions are legal moves. If the rules make no distinction between the players, that
is if both players have the same options of moving from each position, the game is called
impartial; otherwise, the game is called partizan.
The players alternate moving.
The game ends when a postion is reached from which no moves are possible for the
player whose turn is to move.
The game ends in a finite number of moves no matter how it is played.
The Game of Nim
The most famous take-away game is the game of
Nim, played as follows. There are three piles of chips
containing x1, x2, and x3 chips respectively. Two
players take turns moving. Each move consists of
selecting one of the piles and removing chips from it.
You may not remove chips from more than one pile in
one turn, but from the pile you selected you may
remove as many chips as desired, from one chip to
the whole pile. The winner is the player who removes
the last chip.
P-positions and N-positions
Winning for the Previous player
Winning for the Next player to move
Characteristic Property:
(1)All terminal positions are P-positions.
(2) From every N-position, there is at least one move to a Pposition
(3) From every P-position, every move is to an N-position.
Algorithm
Recursion
Step 1: Label every terminal position as a P-position.
Step 2: Label every position that can reach a labelled Pposition in one move as an N-position.
Step 3: Find those positions whose only moves are to
labelled N-positions; label such positions as P-positions.
Step 4: If no new P-positions were found in step 3, stop;
otherwise return to step 2.
Nim-Sum
a1,a2,a3,……an
Nim-Sum=a1^a2^...^an
Theorem of C.L. Bouton
A position, (x1, x2, x3,...,xn), in Nim is a P-position if
and only if the nim-sum of its components is zero,
x1^x2^x3^...^xn=0.
Proof
All terminal positions are in P.
From each position in N, there is a move to a
position in P
Every move from a position in P is to a
position in N
Graph Games
A two-person game played on a directed graph with no cycles
a starting position x0 ∈ X and using the following rules:
(1) Player I moves first, starting at x0.
(2) Players alternate moves.
(3) At position x, the player whose turn it is to move chooses a position
y ∈ F(x).
(4) The player who is confronted with a terminal position at his turn, and
thus cannot move, loses
Minimal Excludant
the smallest non-negative integer not in the set
mex{0,1,2,4}=3;
mex{}=0;
The Sprague-Grundy Function
g(x) = mex{g(y) : y ∈ F(x)}
Character
If x is a terminal position, g(x) = 0.
At postions x for which g(x)=0, every follower y of x is such
that g(y)!=0
At postions x for which g(x)!=0,there is at least one
follower
y such that g(y)=0
The Sprague-Grundy Theorem
A position, (x1, x2, x3,...,xn), is a P-position if and only if
G(x1)^G(x2)^G(x3)^...^G(xn)=0
Examples
poj1704,2425,2311
Reference
• 《GAME THEORY》 Thomas S. Ferguson
• 《超高端桌游：一点一线烧糊你的大脑》 junglerubik
• Wikipedia
Thanks!