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Mo
is Game Variant
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a0 d0 g0 b1 d1 f1 c2 e2 b3 c3 e3 f3 g3 c4 d4 e4 b5 d5 f5 a6 d6 g6






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a b c d e f g
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a b c d e f g
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a0 d0 g0
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c4 d4 e4
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a6 d6 g6
Nine Men’s Mo
is
Nine Men’s Mo
is is a board game between two players: White and Black. There are many online imple-
mentations available online. See, e.g., first link, or second link.
The Mo
is Game Variant is a variant of Nine Men’s Mo
is game. Each player has 9 pieces, and the game
oard is as shown above. Pieces can be placed on intersections of lines. (There are a total of 22 locations
for pieces.) The goal is to remove opponent’s pieces by getting three pieces on a single line (a mill). The
winner is the first player to reduce the opponent to only 2 pieces, or block the opponent from any furthe
moves. The game has three distinct phases: opening, midgame, and endgame.
Opening: Players take turns placing 9 pieces - one at a time - on any vacant board intersection spot.
Midgame: Players take turns moving one piece along a board line to any adjacent vacant spot.
Endgame: A player down to only three pieces may move a piece to any open spot, not just an adjacent
one (hopping).
Mills: At any stage if a player gets three of their pieces on the same straight board line (a mill), then one
of the opponent’s isolated pieces is removed from the board. An isolated piece is a piece that is not part of
a mill.
1
https:
www.mathplayground.com/logic_nine_mens_mo
is.html
http:
ninemensmo
is.ist.tugraz.at:8080
A computer program that plays Variant
The basic components of a computer program that plays Variant are a procedure that generates moves, a
function for assigning static estimation value for a given position, and a MiniMax or AlphaBeta procedure.
Representing board positions
One way of representing a board position is by an a
ay of length 22 , containing the pieces as the letters
W,B, x. (The letter x stands for a “non-piece”.) The a
ay specifies the pieces starting from bottom-left
and continuing left-right bottom up. Here are a two examples:
xxxxxBxWWWWBBBBxxxxxxx WxWWxWWWWBBBBBBBBxxxxx






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a b c d e f g
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1
2
3
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5
6
B
W
W W W B B
B B






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a b c d e f g
0
1
2
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W W
W W
W W
W B B B B
B B B
B
Move generato
A move generator gets as input a board position and returns as output a list of board positions that can be
eached from the input position. In the next section we describe a pseudo-code that can be used as a move
generator for White. A move generator for Black can be obtained by the following procedure:
Input: a board position b.
Output: a list L of all positions reachable by a black move.
1. compute the board tempb by swapping the colors in b. Replace each W by a B, and each B by a W.
2. Generate L containing all positions reachable from tempb by a white move.
3. Swap colors in all board positions in L, replacing W with B and B with W.
A move generator for White
A pseudo-code is given for the following move generators: GenerateAdd, generates moves created by adding
a white piece (to be used in the opening). GenerateMove, generates moves created by moving a white
piece to an adjacent location (to be used in the midgame). GenerateHopping, generates moves created
y white pieces hopping (to be used in the endgame). These routines get as an input a board and generate
as output a list L containing the generated positions. They require a method of generating moves created
y removing a black piece from the board. We name it GenerateRemove.
2
GenerateMovesOpening
Input: a board position
Output: a list L of board positions
Return the list produced by GenerateAdd applied to the board.
GenerateMovesMidgameEndgame
Input: a board position
Output: a list L of board positions
if the board has 3 white pieces Return the list produced by GenerateHopping
applied to the board. Otherwise return the list produced by GenerateMove
applied to the board.
GenerateAdd
Input: a board position
Output: a list L of board positions
L = empty list
for each location in board:
if board[location] == empty {
= copy of board; b[location] = W
if closeMill(location, b) generateRemove(b, L)
else add b to L
}
eturn L
GenerateHopping
Input: a board position
Output: a list L of board positions
L = empty list
for each location α in board
if board[α] == W {
for each location β in board
if board[β] == empty {
= copy of board; b[α] = empty; b[β] = W
if closeMill(β, b) generateRemove(b, L)
else add b to L
}
}
eturn L
3
GenerateMove
Input: a board position
Output: a list L of board positions
L = empty list
for each location in board
if board[location]==W {
n = list of neighbors of location
for each j in n
if board[j] == empty {
= copy of board; b[location] = empty; b[j]=W
if closeMill(j, b) GenerateRemove(b, L)
else add b to L
}
}
eturn L
GenerateRemove
Input: a board position and a list L
Output: positions are added to L by removing black pieces
for each location in board:
if board[location]==B {
if not closeMill(location, board) {
= copy of board; b[location] = empty
add b to L
} }
If no positions were added (all black pieces are in mills) add b to L.
neighbors and closeMill
The proposed coding of the methods neighbors and closeMill is by “
ute force”. The idea is as follows.
neighbors
Input: a location j in the a
ay representing the board
Output: a list of locations in the a
ay co
esponding to j’s neighbors
switch(j) {
case j==0 (a0) : return [1,3,16]. (These are d0,b1,a6.)
case j==1 (d0) : return [0,4,2]. (These are a0,d1,g0.)
etc.
}
4
closeMill
Input: a location j in the a
ay representing the board and the board
Output: true if the move to j closes a mill
C = b[j]; C must be either W or B. Cannot be x.
switch(j) {
case j==0 (a0) : return true if
(b[1]==C and b[2]==C)
or (b[3]==C and b[6]==C)
else return false
case j==1 (d0) : return true if
(b[0]==C and b[2]==C)
else return false
etc.
}
Static estimation
The following static estimation functions are proposed. Given a board position b compute:
numWhitePieces = the number of white pieces in b.
numBlackPieces = the number of black pieces in b.
L = the MidgameEndgame positions generated from b by a black move.
numBlackMoves = the number of board positions in L.
A static estimation for MidgameEndgame:
if (numBlackPieces ≤ 2) return(10000)
else if (numWhitePieces ≤ 2) return(-10000)
else if (numBlackMoves==0) return(10000)
else return XXXXXXXXXXnumWhitePieces − numBlackPieces) - numBlackMoves)
A static estimation for Opening:
eturn (numWhitePieces − numBlackPieces)
5

Programming project
Artificial Intelligence
The project has 4 parts. In each part you are asked to write two programs. The programming language
should be JAVA, Python, C/C++.
Part I: MINIMAX (45%)
Write two programs that get as input two file names for input and output board positions, and the depth
of the tree that needs to be searched. The programs print a board position after White plays its best move,
as determined by a MINIMAX search tree of the given depth and the static estimation function given in
the Mo
is-Variant handout. That board position should also be written into the output file. In addition,
the programs prints the number of positions evaluated by the static estimation function and the MINIMAX
estimate for that move. The board position is given by a list of 22 letters. See the Mo
is-Variant handout
for additional information.
First program: MiniMaxOpening
The first program plays a move in the opening phase of the game. We request that you name it MiniMax-
Opening.
For example, the input can be:
(you type:)
MiniMaxOpening board1.txt board2.txt 2
(the program replies:)
Board Position: xxxxxxxWxxWxxxBxxxxxx
Positions evaluated by static estimation: 9.
MINIMAX estimate: 9987.
Here it is assumed that the file board1.txt exists and its content is:
xxxxxxxWxxxxxxBxxxxxx
The file board2.txt is created by the program, and its content is:
xxxxxxxWxxWxxxBxxxxxx
(The position and the numbers above are most likely co
ect. They are given just to illustrate the format.)
Please use the move generator and the static estimation function for the opening phase. You are not asked
to verify that the position is, indeed, an opening position. You may also assume that this game never goes
into the midgame phase.
1
Second program: MiniMaxGame
The second program plays in the midgame/endgame phase. We request that you call it MiniMaxGame.
For example, the input can be:
(you type:)
MiniMaxGame board3.txt board4.txt 3
(the program replies:)
Board Position: xxxxxxWWWxWWxBBBBxxxxx.
Positions evaluated by static estimation: 125.
MINIMAX estimate: 9987.
Here it is assumed that the file board3.txt exists and its content is:
xxxxxxxxWWxWWxBBBxxxxx
The file board4.txt is created by the program, and its content is:
xxxxxxWWWxWWxBBBBxxxxx
(The position and the numbers above may not be co
ect. They are given just to illustrate the format.)
Part II: ALPHA-BETA (35%)
In this part you are asked to write two program that behave exactly the same as the program of Part I,
ut implement the ALPHA-BETA pruning algorithm instead of the MINIMAX. Notice that these programs
should return the exact same estimate values as the programs of Part I; the main difference is in the numbe
of nodes that were evaluated. We request that you call these programs ABOpening and ABGame.
Part III: PLAY A GAME FOR BLACK(10%)
Write the same programs as in Part I, but the computed move should be Black’s move instead of White’s
move. We request that you call these programs MiniMaxOpeningBlack and MiniMaxGameBlack.
Part IV: STATIC ESTIMATION (10%)
Write an improved static estimation function. The new function should be better than the one which was
suggested in the handout. Rewrite the programs of Part I with your improved static estimation function.
We request that you call these programs MiniMaxOpeningImproved and MiniMaxGameImproved.
Due date: to be announced.
For C/C++ implementations and other special cases it may be necessary for you to be present when you
project is being tested.
What you need to submit:
Submit a documented source code, and, if relevant, executables. This should include the source for all eight
programs.
Show examples of the program output when applied to several positions. Give at least two cases in which
alpha-beta produces savings over MINIMAX.
Show at least two examples where your static evaluation function produces different moves than the standard
evaluation function. Write a short (one or two paragraphs) explanation of why you believe your function to
e an improvement over the function proposed by the instructor.
The submission should be a single zip file named with your net ID. For example, if your net ID is xyz1234
your submission file should be named xyz1234.zip.
2
Community Standards and Conduct
This is an individual project. You may discuss with other students the performance of your program, but
you are not allowed to share code. All programs will be tested for plagiarism.
3
Answered 4 days After Mar 29, 2023

Solution

Aditi answered on Apr 01 2023
26 Votes
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