Checkers is a Game Played on an Eight by Eight Squared Board With Twelve Pieces on Each Side

8/6/2019 Checkers is a Game Played on an Eight by Eight Squared Board With Twelve Pieces on Each Side

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Index

Preface 3

Introduction 4

History 5

Levels of solving a game 6

Solving checkers 8

Implementation 14

Demonstration 15

Screen shots 19

Conclusion 20

References 21

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PREFACE

Checkers is a game played on an eight by eight squared board with twelve

pieces on each side, each might only initially move and capture diagonally

forwards, and it can move both backwards and forwards only when the piece

is Kinged.

As long as the checkers can be unbeatable game, Arthur Samuel thought

about using the Lees guide to Checkers for creating a learning game that

will adjust the criteria for choosing the moves, The program was able to learn

from its errors and achieved an impressive playing strength. It was capable

of beating the world champion of checkers at that time.

As a hint from the book Checkers is Solved By Jonathan Schaeffer, Neil

Burch, Yngvi Bjrnsson, Akihiro Kishimoto, Martin Mller, Robert Lake, Paul

Lu, Steve Sutphen. Science (4 September 2007: Vol. 317. no. 5844, pp. 1518

- 1522): "The game of checkers has roughly 500 billion billion possible

positions (5 x 1020). The task of solving the game, determining the final

result in a game with no mistakes made by either player, is daunting. Since

1989, almost continuously, dozens of computers have been working on

solving checkers, applying state-of-the-art artificial intelligence techniques to

the proving process. This paper announces that checkers is now solved:

Perfect play by both sides leads to a draw. This is the most challenging

popular game to be solved to date, roughly one million times as complex as

Connect Four. Artificial intelligence technology has been used to generate

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Introduction

Its not easy to solve a game, specially that the result to be shown in the

final of the game has to be with no mistake depending on any move that can

be done by the player, any thought should be considered in checkers

specially that there are assumptions that have been made about any step a

play can do with a piece, and these assumptions are not just about the next

move but they are about every move that can be done during the whole

game period from start point till the end point including when the piecebecomes a king and can move backwards and forwards.

Many computers have been used to solve the game applying state-of-the-art

artificial intelligence techniques to the proving process and many others

have been participating in competitions against real human players, thats

why it is always respected to know or make an assumption of what next

move a player can have.

This presentation describes the way of solving the game and it is a

description of a start of designing an unbeatable game that can make all

assumptions that have been mentioned,

This game was already made by many researchers, even though it is good to

study what have been made, to implement what have been created and to

know what have been produced.

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History

It is a board game that in the UK they call it draughts which is one of the

oldest games ever known to man in Europe, once tracing its history we can

go back to the cradle of civilization where vestiges of the earliest form of

the game was unearthed in an archeological dig in the ancient city of Ur in

southern Mesopotamia, which is now modern day Iraq. no one is sure of the

exact rules of the game which was carbon dated at 3000 B.C. A similar game

using a 5x5 board, called Alquerque is known to have existed in ancientEgypt as far back as 1400 B.C.

Checkers (known as draughts in the United Kingdom and some other

countries), also called American checkers, straight checkers, or simply

draught, is a form of draughts board game played on an eight by eight

squared board (with sixty-four total squares) with twelve pieces on each

side. These pieces may only initially move and capture diagonally forwards.

Only when a piece is "crowned" or "kinged" may it move both backwards or

forwards.

As in all draughts variants, English draughts is played by two people, on

opposite sides of a playing board, alternating moves. Traditionally the pieces

are either black, white, or red, and the board alternates between red and

white squares. The opponent's pieces are captured by jumping over them.

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http://en.wikipedia.org/wiki/United_Kingdomhttp://en.wikipedia.org/wiki/Draughtshttp://en.wikipedia.org/wiki/Board_gamehttp://en.wikipedia.org/wiki/United_Kingdomhttp://en.wikipedia.org/wiki/Draughtshttp://en.wikipedia.org/wiki/Board_game


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Levels of solving a game

It is not easy solving a game, especially that there are many steps to be

done during the playing period, each step might have hundreds of

assumptions, each step the human player can think about should be put as

an assumption and that is not easy since its kind of impossible to get into

the human mind, even though its not impossible to think about what a

human mind might think, except in order to solve a game there are levels of

solving it , these levels can be put as following:

Ultra-Weakly solving: The game-theoretic value for the initial

position has been determined.

Weakly solved: The game is ultra-weakly solved and a strategy exists

for

achieving the game-theoretic value from the opening position,

assuming reasonable

computer resources.

Strongly solved : For all possible positions, a strategy is known for

determining the game theoretic value for both players, assuming

reasonable computing resources.

Given the rules of any two-person game with a finite number of

positions, one can always trivially construct a minimax algorithm that

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http://en.wikipedia.org/wiki/Minimaxhttp://en.wikipedia.org/wiki/Minimax


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would exhaustively traverse the game tree . However, since for many

non-trivial games such an algorithm would require an infeasible

amount of time to generate a move in a given position, a game is not

considered to be solved weakly or strongly unless the algorithm can be

run by existing hardware in a reasonable time. Many algorithms rely on

a huge pre-generated database, and are effectively nothing more than

that.

As an example, the game of tic-tac-toe is solvable as a draw for both

players with perfect play (a result even manually determinable by

schoolchildren). Games like nim also admit of a rigorous analysis using

combinatorial game theory .

Whether a game is solved is not necessarily the same as whether it

remains interesting for humans to play. Even a strongly solved game

can still be interesting if the solution is too complex to be memorized;

conversely, a weakly solved game may lose its attraction if the winning

strategy is simple enough to remember (e.g. Maharajah and the

Sepoys ). An ultra-weak solution (e.g. Chomp or Hex on a sufficiently

large board) generally does not affect playability. By David Epstein.

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http://en.wikipedia.org/wiki/Game_treehttp://en.wikipedia.org/wiki/Tic-tac-toehttp://en.wikipedia.org/wiki/Nimhttp://en.wikipedia.org/wiki/Combinatorial_game_theoryhttp://en.wikipedia.org/wiki/Maharajah_and_the_Sepoyshttp://en.wikipedia.org/wiki/Maharajah_and_the_Sepoyshttp://en.wikipedia.org/wiki/Chomphttp://en.wikipedia.org/wiki/Hex_(board_game)http://en.wikipedia.org/wiki/Game_treehttp://en.wikipedia.org/wiki/Tic-tac-toehttp://en.wikipedia.org/wiki/Nimhttp://en.wikipedia.org/wiki/Combinatorial_game_theoryhttp://en.wikipedia.org/wiki/Maharajah_and_the_Sepoyshttp://en.wikipedia.org/wiki/Maharajah_and_the_Sepoyshttp://en.wikipedia.org/wiki/Chomphttp://en.wikipedia.org/wiki/Hex_(board_game)


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Solving checkers

Checkers might be a strongly solved game, there have been thoughts

made, calculations has been created about the probabilities each piece

of checkers can have, and after all there have been considerations of

all possible positions.

It took the researcher and programmer Schaeffer almost 16 years to

create his checkers odyssey, he invented CHINOOK which is the first

program to win in the world championship against a human being in

checkers, this program was made in 1989, the endgame database has

been created to trace backwards from the final results of games such

as wins, losses or draws then there was a deep search algorithm made

to make decisions during play based on evaluating all possible

outcomes a certain number of moves ahead.

Schaeffer said: Strongly solving the game or computing all of these

possible positions would have taken decades, but what he did was firstly

creating a database of the endgames building backward from all the possible

wins, losses or draws that the checkers can include, he created also an

algorithm about backward searching that builds the path of situations that

can lead the mentioned endgames to the point where exist 10 game pieces

on the board, the result that he came up with was a database of 39 trillion

positions compressed using homebrew algorithm into an average of 237

gigabytes for an average of 154 positions per byte of data.

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Backward searching algorithm is one thing, the next step to be done is the

forward search technique which is by the way not performed in the way

Chinook because it used a deep search algorithm to make the next move

always by analyzing all possible situations that are one move deep then all

possible situations that are two moves deep and so on till the end.

There is another technique that actually was also made in Chinook, this

technique is called best first which prioritize the searching various positions

and lines of play at a given position in the games where there are several

possible moves that can be made.

As a definition of each technique to be made:

Backward search: the positions at the end are being searched with their

win/loss/draw value determined, the technique used in this algorithm is the

retrograde analysis which has been successfully used for many games, the

algorithm is called backward search because it works backward by starting at

the end of the game and working toward the start, it also enumerates all one

piece positions declaring each value of each position then it goes to the two

piece position which of course leads also to a one piece position with a known

value the again the three piece positions and so on, the endgame databases

are crucial to solving checkers which forced to capture rule quickly results in

many pieces being removed from the board giving rise to a position with less

than 10 pieces and a known value.

Forward search: consists of two parts: the proof tree manager that builds

the proof by defining the positions to be assessed and the proof solvers which

search individual positions, the manager maintains the master copy of the

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proof and uses the proof number search algorithm to identify a prioritized list

of positions that need to be examined, typically several hundred positions of

interest are generated at a time so as to keep multiple computers busy.

As the checkers board, each square where the pieces can take a move was

converted into a number to be used in algorithms and diagrams.

Respecting the above board assumptions was made and number of

positions was declared as following:

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The number of positions in the game of checkers. For example, the

possible positions for one piece include 32 squares for the Black king,

32 squares for the White king, 28 quares for a Black checker, and 28

squares for a White checker, for a total of 120 positions.

Forward and backward search. The number of pieces on the board are

plotted vertically) versus the logarithm of the number of positions

(Table 1). The shaded area shows the endgame database part of the

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proofi.e., all positions with !10 pieces. The inner oval area shows that

only a portion of the search space is relevant to the proof. Positions

may be irrelevant because they are unreachable or are not required for

the proof. The small open circles indicate positions with more than 10

pieces for which a value has been proven by a solver. The dotted line

shows the Boundary between the top of the proof tree that the

manager sees (and stores on disk) and the parts that are computed by

the solvers (and are not saved in order to reduce disk storage needs).

The solid seeded line shows a best sequence of moves.

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Table 2. Openings solved. Shown are the opening moves (using the

standard square number scheme in Fig. 1, bottom), the result, the number of

positions given to the solvers, and the position farthest from the start of the

game that was searched (Max ply). The last two columns give the size and

ply depth of the pruned minimal proof tree. Note that the total does not

match the sum of the 19 openings. The combined tree has some duplicated

nodes, which have been removed when reporting the total.

Fig. 3. The first three moves of the checkers proof tree. Move sequences are

indicated using the

notation from Fig. 1B, with the from-square and to-square of the move

separated by a hyphen. The result of each position is given for Black, the first

player to move (=D, a proven draw; =L, a proven loss; =D, draw or win). In some positions, only one move needs to be

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considered; the rest are cut off, as indicated by the rotated T. Some

positions have only one legal move because of the forced-capture rule.

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Implementation

The implementation of the checkers should respect the table of

positions and should demonstrate:

Alpha Beta Pruning :

technique for greatly reducing the number of moves searched to get

the exact same result.

it is possible to reach up to almost twice the search depth (if optimal,)

by searching the same number of positions. It is optimal if any time a

player has one or more moves good enough to cause a cut-off, one of

these moves is searched first. Therefore to get the most out of alpha-

beta, you have to try to predict the best move for the player to move,

and try this move first.

Iterative Deepening :

starting with shallow searches before running a deeper search.

Transposition Table (Hash Table):

It is a table in memory that stores information about board positions

previously searched. A plain Alpha-Beta search function will search any

board it receives, but this is wasteful if it has already encountered and

searched this board before in another branch of the search.

Quiescence Search

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It is a necessary in checkers, where the evaluation is based greatly on

material pieces. A quiescence search means searching certain board

positions further before evaluating the board.

Demonstration

I did not create a game that plays against computer because it takes a

long time to do so, but I did create a game that whenever you touch a

piece it gives you the positions that can be done next.

This is a code from the main class of the java game:

import java.awt.*;

import java.awt.event.*;

import javax.swing.*;

public class draughts extends JPanel implements ActionListener

{

private windowDraughtsBoard draughtsBoard;

private objCreateAppletImage createImage;

private JButton cmdNewGame;

private JPanel panBoard, panStatusArea, panButton;

private Color clrBackgrounds;

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public static void main(String[] args)

{

JFrame.setDefaultLookAndFeelDecorated(true);

JFrame frame = new JFrame("Draughts"); //Title

frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

JComponent paneMain = new draughts();

paneMain.setOpaque(true);

paneMain.setPreferredSize(new Dimension(550,

600));

frame.setContentPane(paneMain);

frame.pack();

frame.setVisible(true);

}

public draughts()

{

super.setLayout(new BorderLayout());

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createImage = new objCreateAppletImage();

draughtsBoard = new windowDraughtsBoard();

try

{

Image coinRed = createImage.getImage(this,

"coinRed.gif", 500);

Image coinBlue = createImage.getImage(this,

"coinBlue.gif", 500);

Image coinRedKing =

createImage.getImage(this, "coinRedKing.gif", 500);

Image coinBlueKing =

createImage.getImage(this, "coinBlueKing.gif", 500);

draughtsBoard.setupCoins(coinRed, coinBlue,

coinRedKing, coinBlueKing);

}

catch (NullPointerException e)

{

}

draughtsBoard.setSize(new Dimension(500,500));

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cmdNewGame = new JButton("New Game");

cmdNewGame.addActionListener(this);

panBoard = new JPanel();

panButton = new JPanel();

add(panBoard, BorderLayout.NORTH);

panBoard.add(draughtsBoard);

add(panButton, BorderLayout.SOUTH);

panButton.add(cmdNewGame);

clrBackgrounds = new

Color(75,141,221);

setBackground(clrBackgrounds);

panBoard.setBackground(clrBackgrounds);

panButton.setBackground(clrBackgrounds);

}

public void actionPerformed(ActionEvent e)

{

draughtsBoard.resetBoard();

}

}

Screen-shots

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Conclusion

It is not easy to solve a game , there are always many phases to be

followed and many techniques to be done, with the current existing

technology it is possible to weakly solve a game, specially that a

database contains a proof tree for the game can contain many nodes

which is a small number by todays standards, to solve the game

strongly it requires more nodes to be reached in the game.

As mentioned in this paper the results of Chinook demonstrated a

super powers byt defeating a human being, the project was a success

that belongs to both AI and parallel computing.

So the checkers calculations always eliminate the boundaries of what

can be achieved by search intensive algorithms.

Now that the checkers is solved the question remains how solved is it?

all or almost, are the assumptions enough or missing something, as

long as algorithms show up from nowhere , thoughts are developing

everywhere question will be always put.

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References

Uwic Library

J. Schaeffer et al. , Solving Checkers

(www.ijcai.org/papers/0515.pdf, 2005).

J. Schaeffer et al., in Advances in Computer Games, J.

van den Herik, H. Iida, E. Heinz, Eds. (Kluwer,

Dordrecht, Netherlands, 2003).

Wikipedia

Longest 10PC MTC

(http://pages.prodigy.net/eyg/Checkers/longest-10pcmtc.

htm, 2007).

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Documents

Checkers is a Game Played on an Eight by Eight Squared Board With Twelve Pieces on Each Side