Phutball
Encyclopedia
Phutball is a two-player board game
described in Elwyn Berlekamp
, John Horton Conway
, and Richard Guy's Winning Ways for your Mathematical Plays
.
In this article the two players are named Ohs (O) and Eks (X).
The board is labeled A through P (omitting I) from left to right and 1 to 19 from bottom to top from Ohs' perspective.
Rows 0 and 20 represent "off the board" beyond rows 1 and 19 respectively.
Given that specialized phutball boards are hard to come by, the game is usually played on a 19×19 Go
board, with a white stone representing the football and black stones representing the men.
The objective is to score goals by using the men (the black stones) to move the football (the white stone) onto or over the opponent's goal line. Ohs tries to move the football to rows 19 or 20 and Eks to rows 1 or 0.
At the start of the game the football is placed on the central point, unless one player gives the other a handicap, in which case the ball starts nearer one player's goal.
Players alternate making moves.
A move is either to add a man to any vacant point on the board or to move the ball.
There is no difference between men played by Ohs and those played by Eks.
The football is moved by a series of jumps over adjacent men.
Each jump is to the first vacant point in a straight line horizontally, vertically, or diagonally over one or more men.
The jumped men are then removed from the board (before any subsequent jump occurs).
This process repeats for as long as there remain men available to be jumped and the player desires. Jumping is optional, there is no requirement to jump.
Note that in contrast to checkers, multiple men in a row are jumped and removed as a group.
The diagram on the right illustrates a jump.
If the football ends the move on or over the opponent's goal line then a goal has been scored.
If the football passes through your goal line, but ends up elsewhere due to further jumps, the game continues.
The game is sufficiently complex that checking whether there is a win in one (on an m×n board) is NP-complete
. It is not known whether any player has a winning strategy or both players have a drawing strategy.
Board game
A board game is a game which involves counters or pieces being moved on a pre-marked surface or "board", according to a set of rules. Games may be based on pure strategy, chance or a mixture of the two, and usually have a goal which a player aims to achieve...
described in Elwyn Berlekamp
Elwyn Berlekamp
Elwyn Ralph Berlekamp is an American mathematician. He is a professor emeritus of mathematics and EECS at the University of California, Berkeley. Berlekamp is known for his work in information theory and combinatorial game theory....
, John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
, and Richard Guy's Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games...
.
Rules
Phutball is played on the intersections of a 19×15 grid using one white stone and as many black stones as needed.In this article the two players are named Ohs (O) and Eks (X).
The board is labeled A through P (omitting I) from left to right and 1 to 19 from bottom to top from Ohs' perspective.
Rows 0 and 20 represent "off the board" beyond rows 1 and 19 respectively.
Given that specialized phutball boards are hard to come by, the game is usually played on a 19×19 Go
Go (board game)
Go , is an ancient board game for two players that originated in China more than 2,000 years ago...
board, with a white stone representing the football and black stones representing the men.
The objective is to score goals by using the men (the black stones) to move the football (the white stone) onto or over the opponent's goal line. Ohs tries to move the football to rows 19 or 20 and Eks to rows 1 or 0.
At the start of the game the football is placed on the central point, unless one player gives the other a handicap, in which case the ball starts nearer one player's goal.
Players alternate making moves.
A move is either to add a man to any vacant point on the board or to move the ball.
There is no difference between men played by Ohs and those played by Eks.
The football is moved by a series of jumps over adjacent men.
Each jump is to the first vacant point in a straight line horizontally, vertically, or diagonally over one or more men.
The jumped men are then removed from the board (before any subsequent jump occurs).
This process repeats for as long as there remain men available to be jumped and the player desires. Jumping is optional, there is no requirement to jump.
Note that in contrast to checkers, multiple men in a row are jumped and removed as a group.
The diagram on the right illustrates a jump.
- Ohs moves the football from K6-G9-G11-J11.
- The men on J7, H8, G10, and H11 are removed.
- The jump from K6-G9-J9-G7 would not be legal, as that would jump the man on H8 twice.
If the football ends the move on or over the opponent's goal line then a goal has been scored.
If the football passes through your goal line, but ends up elsewhere due to further jumps, the game continues.
Strategy
- Carefully set up sequences of jumps can be "spoiled" by extending them at critical moments.
- A jump to the left or right edge can be blocked by leaving no vacant points.
- When you jump, it is usually bad to leave an easily used return path for your opponent to "undo" your progress.
The game is sufficiently complex that checking whether there is a win in one (on an m×n board) is NP-complete
NP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
. It is not known whether any player has a winning strategy or both players have a drawing strategy.
External links
- Dariusz Dereniowski: Phutball is PSPACE-hard
- The game can be played on Sensei's Library and on Vying Games.
- Web Page on Richard Rognlie's Play-By-eMail Server Current and archived games can be seen here
- igGameCenter Play Phutball online in a browser or with an iGoogle gadget
- ConwayGo - Phutball Program in Java