Problem Decomposition: Goal Trees, Rule Based Systems, Rule Based Expert Systems. Planning: STRIPS, Forward and Backward State Space Planning, Goal Stack Planning, Plan Space Planning, A Unified Framework For Planning. Constraint Satisfaction : N-Queens, Constraint Propagation, Scene Labeling, Higher order and Directional Consistencies, Backtracking and Look ahead Strategies.

1. Topic To Be Covered:
Example of CSP:N-Queens Problem Solution
Using Backtracking
Jagdamba Education Society's
SND College of Engineering & Research Centre
Department of Computer Engineering
SUBJECT: Artificial Intelligence & Robotics
Lecture No-10(UNIT-02)
Prof.Dhakane Vikas N

2. N-Queens Problem Solution Using Backtracking
What is N-Queen Problem?
 This problem is to find an arrangement of N queens on a chess board,
such that no queen can attack any other queens on the board.
 The N Queen is the problem of placing N chess queens on an N×N
chessboard so that no two queens attack each other.
 Queen can attack each other if they are in same column, row & diagonal.
 Problem Statement: We have to place 4-Queens on 4 x 4 chessboard such
that no queen will attack each other(For that we have to check no two
queens are placed in same Row, Column & Diagonal

3. N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-
Queens
Step:-1
r=0,c=0
Check, board[0][0] =safe
Now ,Increment column By 1(Place Queen)
Step:2
r=0,c=1
Check, board[0][1] ≠ safe
Now ,Increment Row By 1
Step:3
r=1,c=1
Check, board[1][1] ≠ safe
Now ,Increment Row By 1
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4. N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-
Queens
Step:-4
r=2,c=1
Check, board[2][1] =safe
Now ,Increment column By 1(Place Queen)
Step:5
r=0,c=2
Check, board[0][2] ≠ safe
Now ,Increment Row By 1
Step:6
r=1,c=2
Check, board[1][2] ≠ safe
Now ,Increment Row By 1
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5. N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-Queens
Step:-7
r=2,c=2
Check, board[2][2] ≠ safe
Now ,Increment Row By 1
Step:8
r=3,c=2
Check, board[3][2] ≠ safe
Now ,Increment Row By 1
Now after this if we try to increment Row by 1 it
becomes 4..which is invalid, as we have 4*4
chessboard here, so here we have reach to END of
Board.. and we could not able to placed 4 queens so,
here we have to BACKTRACKKKKK…
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6. N-Queens Problem Solution Using Backtracking
Example :4 x 4 Chessboard To place 4-Queens
 Note: When u backtrack we have to remove last
placed queen and increment ROW BY 1
 So ,here at position Board[2][1] we have lastly
placed the queen so ,we remove it increment ROW
BY 1…So we get next step as follows…
Step:9
r=3,c=1
Check, board[3][1] =safe
Now ,Increment Column By 1(placed Queen)
Step:10
r=0,c=2
Check, board[0][2] ≠ safe
Now ,Increment Row By 1
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7. N-Queens Problem Solution Using Backtracking
Step:11
r=1,c=2
Check, board[1][2] =safe
Now ,Increment Column By 1(placed queen)
Step:12
r=0,c=3
Check, board[0][3] ≠ safe
Now ,Increment Row By 1
Step:13
r=1,c=3
Check, board[1][3] ≠ safe
Now ,Increment Row By 1
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8. N-Queens Problem Solution Using Backtracking
Step:14
r=2,c=3
Check, board[2][3] ≠ safe
Now ,Increment Row By 1
Step:15
r=3,c=3
Check, board[3][3] ≠ safe
Now ,Increment Row By 1
 Now after this if we try to increment Row by 1 it
becomes 4..which is invalid, as we have 4*4
chessboard here, so here we have reach to END of
Board.. and we could not able to placed 4 queens
so, here we have to BACKTRACKKKKK…
Q
Q
Q
Q
Q
Q

9. N-Queens Problem Solution Using Backtracking
 Note: When u backtrack we have to remove last
placed queen and increment ROW BY 1
 So ,here at position board[1][2] we have lastly
placed the queen so ,we remove it increment ROW
BY 1…So we get next step as follows…
Step:16
r=2,c=2
Check, board[2][2] ≠ safe
Now ,Increment Row By 1
Step:17
r=3,c=2
Check, board[3][2] ≠ safe
Now ,Increment Row By 1
Q
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10. N-Queens Problem Solution Using Backtracking
 Now after this if we try to increment Row by 1 it becomes 4..which is
invalid, as we have 4*4 chessboard here, so here we have reach to END
of Board.. and we could not able to placed 4 queens so, here we have to
BACKTRACKKKKK…
 Note: When u backtrack we have to remove last placed queen and
increment ROW BY 1
 So ,here at position board[3][1] we have lastly placed the queen so ,we
remove it increment ROW BY 1
 Now after this if we try to increment Row by 1 it becomes 4..which is
invalid, as we have 4*4 chessboard here, so here we have reach to END
of Board.. and we could not able to placed 4 queens so, here we have to
BACKTRACKKKKK…

11. N-Queens Problem Solution Using Backtracking
 Note: When u backtrack we have to remove last
placed queen and increment ROW BY 1
 So ,here at position board[0][0] we have lastly
placed the queen so ,we remove it increment ROW
BY 1…So we get next step as follows…
Step:18
r=1,c=0
Check, board[1][0] = safe
Now ,Increment column By 1
Step:19
r=0,c=1
Check, board[0][1] ≠ safe
Now ,Increment Row By 1
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12. N-Queens Problem Solution Using Backtracking
Step:20
r=1,c=1
Check, board[1][1] ≠ safe
Now ,Increment Row By 1
Step:21
r=2,c=1
Check, board[2][1] ≠ safe
Now ,Increment Row By 1
Step:22
r=3,c=1
Check, board[3][1] = safe
Now ,Increment Column By 1
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13. N-Queens Problem Solution Using Backtracking
Step:23
r=0,c=2
Check, board[0][2] =safe
Now ,Increment Column By 1
Step:24
r=0,c=3
Check, board[3][0] ≠ safe
Now ,Increment Row By 1
Step:25
r=1,c=3
Check, board[1][3] ≠ safe
Now ,Increment Row By 1
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14. N-Queens Problem Solution Using Backtracking
Step:26
r=2,c=3
Check, board[2][3] =safe
Now ,Increment Column By 1
 So, finally we have placed all 4 queen
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