Table of Contents
- Combinatorial Games: 2 players with turns, complete information. Each player has the same set of moves, and there is no randomness. There are no ties.
- is called terminal if .
- Normal Play: Player on whose turn is faced with a terminal position loses.
- Misère Play: Player on terminal position wins.
Combinatorial Game Analysis
To analyze a game, we classify all positions as or as follows:
- : one from where the previous player can guarantee a victory. Note that this means we assume optimal play from both players.
- : one from where the next player can guarantee a victory.
Under Normal Play, all terminal positions are P positions. Conversely, under Misère Play, all terminal positions are N positions.
These are the simplest take-away games. The game begins with chips. On your turn, you take away chips from the pile, where .
Let us analyze the set .
An observation: A position is a position if is divisible by .
This leads us to formulate the following hypothesis:
Proof by induction
We first note that .
Assume that our inductive hypothesis is true .
We show that it is true for .
Recall that we defined the following sets:
Claim: Every is in , and . Read the proof of this in Theorem 1.1.5 in KP.
A winning strategy is a set of moves from that can guarantee a win.
A graph is defined as , where is a set of vertices and is a set of edges connecting the vertices. We can define a graph on our state space, which (upon inspection) is a DAG since all states can only transition into smaller states.
A Recursive Algorithm to Label Positions
(Ferguson) Recursive Algorithm to Label Positions:
- Label all the terminal positions as .
- Label each position that has an edge to a -position as .
- If a position is not labeled yet, then check the edges. If there exists at least one edge to a -position, label this position . Otherwise all edges lead to -positions, so we label this position .
Analyze , and give a general rule.
Invented by David Gale. Related to divisor game (Frederik Schuh). Chomp.