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-------------------------------------------------------------------------
THE RULES OF OTHELLO
"Othello - A minute to learn, a lifetime to master!" - The game of
Othello is derived from an ancient game called Reversi, which started on
the same 8x8 board as Othello, but with none of the center squares
occupied. The rules of Reversi stated that the first four moves had to
take place in the four center squares, thereby giving rise to one of two
starting formations, a parallel formation and a checkered center
formation. The checkered center formation as a starting point was
copyrighted into the game of Othello which after various ownerships under
Gabriel and Milton Bradley, currently rests with Pressman Toy Company.
The Othello rules are simple. Black moves first and the moves alternate
between players. A player must move as long as he has at least one legal
move, else he passes. A legal move consists of any move that "flips" at
least one of the opposing pieces. An opposing piece is flipped if it
stands between (either alone or with a line of other opposing pieces and
either vertically, horizontally, or diagonally) the piece just played and
another piece of the same color as that just played. When a piece is
flipped, it becomes the opposite color to what it was prior to flipping.
The game ends when neither player has a legal move. The winner of the
game is the player with the most pieces of his color at the end of the
game.
-------------------------------------------------------------------------
REVERSI 4.0 - PROGRAM OPTIONS
Novice Level - This level consists of a random factor which plays a
different game every time. The Novice Level will
generally make more good moves than bad moves, but will
occassionally make some really bad moves. This level is
good practice for true beginners. The level plays a
perfect endgame only 3 moves from the end.
Advanced - This level also consists of a random factor but it is
somewhat limited so that good moves are played much more
often than bad moves. Still, an occasional truly bad
move is not unusual. This level plays a perfect endgame
6 moves from the end (with 6 empty squares left).
Master - The toughest of all levels, there is no random play here.
The Master Level always makes the best play, consistent
with the number of half-moves ("ply") of look ahead which
has been assigned. Based on play against other top
players and programs, it is my belief that the Master
Level at a 3 ply search would be ranked among the top
30 U.S. players. Perfect endgame play occurs 9 moves
from the end at the 3 ply level, 10 moves at the 4 ply
level, etc..
In addition to the above features, the Master level
employs an opening book which has been developed from
world class games. This opening book is available ONLY
at the Master Level.
Setup Mode - The Setup Mode allows you to set up any position and
have the computer start to play from that position.
Once in Setup Mode, position the cursor. Type "B" for
a Black piece on the cursored square, "W" for a White
piece on the cursored square, or hit the space bar for
a blank square. When preparing the position, remember
that you will move first prior to the computer. Your
piece color is established by the color selection menu.
After setting up the correct position, hit the <ENTER>
key. You will then be asked to enter your move, and
the game is underway.
New Game - To start the game with the original Othello position,
select this mode option.
White Pieces - Select this option to play the White pieces, moving second
from the original Othello position.
Black Pieces - Select this option to play the Black pieces, moving first
from the original Othello position.
Let's Play! - Select this option to start the game.
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MASTER LEVEL OPTIONS
Master Level Options are available only when the Master Level is
selected.
2 Ply - A "ply" is the number of half-moves
3 Ply (one move by one player is considered a half-move)
4 Ply of look-ahead for which the program will search.
5 Ply Approximate times of play are given for
6 Ply the IBM XT (8086 with 8 Mhz speed) and for
the IBM AT (80286 with 12 Mhz speed)
Display Option - The computer's thoughts will be displayed as it
considers moves. Only the best move found will
be displayed. The display consists of the value
of the position (positive numbers mean the computer
feels it is winning, negative numbers mean that
it thinks it is losing), and the row and column
numbers of the best variation. Rows and columns
are numbered as follows:
1 2 3 4 5 6 7 8
1 - - - - - - - -
2 - - - - - - - -
3 - - - - - - - -
4 - - - O x - - -
5 - - - x x x - -
6 - - - - - - - -
7 - - - - - - - -
8 - - - - - - - -
The above shows the position after a first move
by Black (x) of row 5, column 6.
No Display - Choose this option if you don't want to see the
computer's thoughts.
------------------------------------------------------------------------
AN INTRODUCTION TO OTHELLO STRATEGY
Welcome to the World of Othello!
Reversi 4.0 is the result of alot of hard work, spent not only in the
actual programming but also in learning the mathematics of gaming theory
(from an article on alpha-beta pruning by D.E. Knuth and R.W. Moore) and,
most importantly, in learning the correct strategies in the game of
Othello. When I first considered writing the program in March of 1989, I
had spent about a year playing serious tournament Othello on the USA
Today Sports Center online telecommunications network and was concerned
about the lack of correct strategy in most of the Othello programs on the
market. While Othello is simple to learn it is indeed complex to master,
especially without good opponents to play. Thus, my intention was to
create a good opponent for interested Othello players.
In addition to a lack of good strategical programs, there is also a lack
of good instructional text written in English. The only book I had ever
read on Othello, written by the Japanese inventor of the game, contained
incorrect and often very bad strategical advice. The best instructional
text (and in fact the ONLY good text written in English of which I am
currently aware) is contained in the quarterly magazine of the United
States Othello Association (USOA) and in the magazine of the British
Othello organization. Through these magazines and through the wonderful
data base program, OTHCGA, written by Brian Rose (one of America's top
two players) I was able to open the doors to the truly correct Othello
strategy.
Through these last few years I've met new Othello friends, learned the
basics of Othello strategy, and had enough success in national Othello
tournaments to achieve a United States ranking among the top ten Othello
players.
In July of 1989 I distributed my first program, Reversi 2.0, through the
shareware concept. The purpose of such a distribution was not to make
money (and indeed, I received only about 10 contributions totalling $150)
but rather to publicize the game of Othello in the hopes of attracting
new players to the game.
The original version of this game is nowhere near as strong as the
current version, Reversi 4.0, and it occurred to me that rather than
attract players to the game, the new version might scare people away from
it. In fact, the current version of the game, set on the master level at
3 ply of thought, is probably good enough to routinely beat all but the
top 30 or 40 players in the country.
To give the beginning player some help with the fundamentals of Othello,
I've decided to include here an elaboration of all of the concepts which
have been employed in the Reversi 4.0 program. Unlike chess, Othello is
in its infancy and the concepts are continuing to be refined and
developed. Also unlike chess, it is possible to be a top Othello player
in a short period of time. Indeed, a few years ago a French player by
the name of Ralle won the world Othello championship just a few years
after learning the game.
It should be stated here that Othello is not a "solved" game. The best
players in the world are not computers, despite some very extensive
programming efforts by some very fine programmers. It should also be
stated here that, while the second player to move has a slight advantage
(because of the parity concept described later), this advantage is not
overwhelming. Indeed, as in chess, the very best players of Othello are
overwhelming favorites against very good players. The top two players in
the United States (at this time) are expected to win 19 out of 20 games
against the third ranked U.S. player.
Herein, then, lie the concepts I've learned and employed in my program.
THE FUNDAMENTAL STRATEGIES OF OTHELLO
I. Specific Square Values - In the only English speaking book on Othello
that I've ever seen, the inventor of the game put forth the theory
that certain squares were more attractive to control than other
squares. While this theory is completely ill-founded the names that
he gave to certain squares stuck in the Othello world. The following
diagram indicates the squares he named (a "-" represents an unnamed
square):
- C A B B A C -
C X - - - - X C
A - - - - - - A
B - - - - - - B
B - - - - - - B
A - - - - - - A
C X - - - - X C
- C A B B A C -
Of the squares listed in the above diagram, only three types merit
specific attention and these are the corner squares, the "X" squares,
and the "C" squares.
One strategy that the beginner learns very early is that it is very
important to occupy the corner square and to keep your opponent from
occupying the corner square. The reason for this is that the corner
square is absolutely stable, i.e. it can not be flipped regardless of
the board position. Once a corner square is occupied it never changes
color until the end of the game.
Because the corner square is such an important square to occupy, the
"X" square is important NOT to occupy, because it gives the opponent
opportunities to enter the corner square. There are some instances
where an "X" square occupation is correct, but these generally involve
the sacrifice of one corner to the opponent for the opportunity to
occupy a different corner.
The "C" square would at first seem as dangerous as the "X" square to
occupy, since it is one square away from the corner and therefore
provides a means of the opponent's occupation of the corner. However,
the entrance to the corner by means of the "C" square can only occur
from the sides. It is true that the "C" square is not a good square
to occupy, but it may not be entirely bad either. The correctness
or incorrectness of playing into the "C" square depends entirely on
the types of formations occuring along the edge containing the
"C" square. We will discuss these formations later.
In my program, I give a marginally bad value to a move to the "C"
square. The value is much worse to an "X" square play. Finally, I
give a highly positive value to the occupation of a corner square.
II. Side Formations
The occupation of the edges appeals to many starting players,
primarily because the sides have more stability than the middle
squares. Indeed, sides can only be flipped by opponents pieces
occupying squares on the same side. However, in master class play
occupation of the sides tends to be avoided. A statistical analysis
of world class games shows, in fact, that the first player to occupy
an edge tends to lose more games than the player who is not first.
Occupation of the sides has one advantage, that of stability, yet has
the disadvantage of allowing a number of tactical combinations which
have as a result the forced occupation of a corner by the opponent's
piece. It is important to recognize the tactical moves which cause
certain side formations to be dangerous. Most of these tactical
dangers are available only when the "C" square is occupied.
The first type of danger involves the outright win of a corner without
a compensating gain of another corner. The following positions show
only the bottom row of an Othello board. In each case the "O"s can
force the win of a corner from the "x"s, regardless of which side is
on move:
A) - - - O - O x - B) - - O O - O x -
C) - - O - O x x - D) - - O x - O x -
E) - - - - x O x - F) - - - x O x x -
G) - - x O x x x - H) - - x O O x x -
Note that each position involves the dangerous x occupation of the
"C" square. It is important to recognize edge formations which
give rise to some of the above positions:
A) - - - O - - x - B) - - O O - - x -
C) - - O - - x x - D) - - O x - - x -
E) - - - - x - x - F) - - - x - x x -
G) - - x - x x x - G2) - - x - - - x -
Each of the above positions give rise to the associated edge positions
previously described, provided that it is "O"s turn to play and
that "O" has the opportunity to move to the crucial edge squares.
Note especially that edge formations with an odd number of gaps
are very dangerous.
The second type of danger in edge formations comes from the opponent's
sacrifice of one corner for the opponent's occupation of another
corner and of most of the edge extending from that corner. Once a
corner is occupied, and therefore totally stable, all pieces extending
directly along the edge of the board from the stable corner are also
totally stable. The second type of danger occurs in the following
edge formations:
1) - x x x x x O x 2) - x x x x O O x
Again, the danger occurs in the following formations when "O" is on
the move and has access to the crucial empty edge squares:
3) - x x x x x - x 4) - x x x - - - x
5) - x x x O - - x 6) - x O - - - - x
7) - x x - O - - x 8) - x - x O - - x
When looking at side formations, it is often important to look
at the parallel row formations adjoining the edge. Consider for
example the following examples of two rows on the south edge of the
board:
i) - O x - - - - - ii) - - x - - - - -
- - x x x x x - - - x x x x x -
iii) - O O - - - - - iv) - - O - - - - -
- - x x x x x - - - x x x x x -
Position (i) is always bad for the x's regardless of who is on
move. If the southwest corner is taken by the x's, the position
in (3) occurs which is favorable for the O's. Because position (ii)
tends to lead to position (i), it is also usually bad. Position
(iii) may be bad only if an x move to the south "C" square flips
the O piece on the southwest "X" square. Finally, position (iv)
is usually not bad since it gives the x's a means of occupying
the important south "C" square.
A very important concept in Othello play was discovered by a U.S.
player named John Stoner. The edge formation trap described below
is named the "Stoner Trap" after him. Consider the following
southern edge positions:
a) - x O - O - - - b) x - O O x O - -
- O x x - - - - - O x x x - - -
- - - x x x x - - x - - x x x -
In each of the above cases, assume that it is x's move and that
x cannot occupy the southwest corner square ON THIS MOVE. In both
cases, after the x move, the O's have a very favorable play which
forces either the win of the southwest corner or the win of the
southeast corner and the south edge majority in exchange for the
sacrifice of the southwest corner to the x's. The threat of springing
these "Stoner Traps" is a common consideration in world class play,
and in the Reversi 4.0 program.
III. Disk Stability and Maximization of Options
One of the classic truths of the game of Othello involves an
instructive position:
- O O O O O O -
O O O O O O O O
O O O O O O O O
O O O O O O O O
O O O O x O O O
O O O O O O O O
O O O O O O O O
- O O O O O O -
While the O's appear to be winning the game, in fact the last
four moves are all x moves which allow x to flip enough disks to
win the game 38-26. Why did the x's win? If you look at the
above position you will note that the x island in the midst of
the O's has two advantages which can be extended to correct Othello
strategy.
The first of these advantages is called Disk Stability, which occurs
when disks cannot be flipped by any opponent move or future opponent
move. Certainly the lone x disk in the above diagram qualifies as a
stable disk.
The second of the advantages involves the maximization of your options
when compared to your opponent's options. In the above diagram, the
O's have no move options at all. It is worthy to note here that it
is just as important in Othello to minimize your opponents move
possibilities as it is to maximize your move options.
The combination of these two concepts gives rise to the correct
analysis that stable cluster formations in the center of the board
are most advantageous (the aforementioned position is an illustration
of this).
A further observation about the combination of these two concepts is
that it is important to keep your disks connected to each other in the
opening. Disks which are in the center of two separate opponent
formations tend to be both stable and also tend to limit the
opponent's options.
The Reversi 4.0 program evaluates the stability and option potential
of various disk formations. In addition, a value is employed of the
internal cluster and disk connectivity concepts by the invention of
an "Average Foreign Neighbor" statistic. This statistic in the
program looks at every disk and assigns a weight of 1 for every
direct neighboring square which contains an opponent's disk and a
weight of 2 for every direct neighboring square which is empty.
The program then tries to minimize this statistic.
The Average Foreign Neighbor statistic has a few important effects.
First, it is better for a disk to be near other disks its own color.
Second, it is better to be in the center of opponent's disks than on
the perimeter of any formation (where your disk has many surrounding
empty squares). Third, a disk along the edge has the advantage of
having less neighboring squares and there is a natural affinity for
the edges through this statistic.
IV. Evaporation Strategy
Many a beginner has started the game of Othello by striving in the
opening for what his goal is at the end, trying to maximize the number
of disks he has on the board. Yet, there is not an opening strategy
that could be worse than this. Let us take a look at why.
With respect to the rule of Option Maximization, it is true that the
side with more disks tends to have less possible legal moves. This is
because a legal move requires the flipping of your opponent's pieces.
If your opponent has more pieces to flip, you probably have more legal
moves. If you have more pieces, your opponent probably has more legal
moves.
In fact, a correct Othello opening strategy is that of minimizing your
pieces, also called "Evaporation Strategy." While this strategy is
somewhat less important than the other strategies mentioned above it
is nevertheless very important not to let the number of your pieces
overwhelm the number of your opponent's pieces.
A term coined by one of the top ten U.S. players, Arnold Kling, in an
issue of the USOA newletter, the Othello Quarterly, illustrates the
combined strategies of evaporation and the Average Foreign Neighbor
statistic. The term, the "Perfectly Quiet Move" is a rule that any
move (in the opening or middlegame) that flips exactly one disk and
as a result places only one additional disk on the perimeter of the
formations is in fact a very favorable move to make. Arnold explains
that much of opening play surrounds the creation of Perfectly Quiet
Move opportunities and the elimination of PQM opportunities for the
opponent.
While the Reversi 4.0 program doesn't explicitly evaluate PQM
opportunities, the employment of both the Average Foreign Neighbor
statistics and of Evaporation Strategy implicitly recognizes PQM.
V. Parity
The most important concept in endgame play (and thus middlegame play
as a precursor to the endgame) is the establishment of move parity.
Consider the following position:
- - O O O O O O
- - O O O O x O
x x x O O O x O
x x x O O O x O
x x x x x O x O
x x x x x O x O
x O O O O O x O
x x x x x x x O
This position shows an empty region in the northwest corner of the
board which has an EVEN number of empty squares. The rule of parity
states that a move into an enclosed region containing an even number
of empty squares is a bad move and, conversely, a move into an
enclosed region containing an odd number of empty squares is a good
move. This rule develops from the fact that the last person to
play into a given enclosed region flips disks which subsequently
become totally stable because there are no more regional flips
available. In the above position, whoever moves first loses the game.
If the region had one more empty square, making five empty squares,
the player to move first would win.
The program, Reversi 4.0, deals with parity in terms of two evaluation
rules. First, the number of empty squares within a given quarter of
the board (quadrant) is counted. Parity becomes more important to
the evaluation routine as the number of empty squares in the quadrant
decreases. Second, the program determines parity advantages by
counting the move opportunies within the empty squares of a quadrant
region. If the region is even, the player with the most opportunities
to play within that region has an disadvantage. If the region is odd
the player with the most opportunities to play within the region has
an advantage. An opportunity is defined in this regard as a count of
the number of squares within the empty region to which a player has
access (to which he has a legal move).
Parity in the endgame is in many instances far more important than the
gaining of corner squares.
The rules and concepts presented here are by no means complete, but
they represent the total of all concepts incorporated in my program.
A studious review of these concepts will be amply rewarded.
While these concepts are all correct, the game of Othello is made
original by the player's judgement as to which of conflicting
objectives is the most important. The program is crudely consistent
in the relative weights of the objectives. Perhaps this is a drawback
of all computer gaming programs, for humans have the flexibility to
tailor the importance of objectives to the position at hand.
ABOUT THE AUTHOR
David W. Parsons resides in New York City with his wife, Marie, and
his three year old son, Christopher. Mr. Parsons is a Vice President
for Metropolitan Life Insurance Company and is a Fellow of the Society
of Actuaries. Besides his ranking as one of the top ten Othello
players in the United States, Mr. Parsons has attained a Master rank
in Duplicate Contract Bridge and a National Master rank in chess,
among the top 300 players in the country.
-------------------------------------------------------------------------
Correspondence regarding this program can be directed to:
David W. Parsons
7 Peter Cooper Road #10G
NY, NY 10010
Phone: (212) 475-8911
Please support Othello by joining the United States Othello
Association and by playing in USOA tournaments. Membership
in the USOA includes the quarterly magazine and is available
for $6 per year. Subscription requests may be sent to:
OTHELLO QUARTERLY
920 Northgate Avenue
Waynesboro, VA 22980
