16-Queen's Problem

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ASSIGNMENT-ARTIFICIAL

INTELLIGENCE

16-QUEENS PROBLEM USING CSP

K.INIYACSE PREFINAL YEAR

The Queen can move any number of vacant squares diagonally, horizontally and vertically.

RULES:

Place 16 Queen’s in 16X16 Chess Board with no interaction between other Queen’s.

Use Constraint Satisfaction Problems (CSPs) Concept to Solve this Problem.

CONT.,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

16X16 CHESS BOARD

PLEASE HELP ME TO FIND

THE RIGHT

SOLUTION

X13

X9

X1

X5

X16

X8

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X15

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X3

X11

X4

X12

X2

X10

CONSTRAINTS

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{3,4,5,6,7,8,9,10,11,12,13

, 14,15,16

}

{5,6,7,8,9,10,11,12,13,

14,15,16} {2,7,8,9,

10,11,12,13,

14,15,16}

{2,4,9, 10,11,12

,13, 14,15,16

}

{4,11, 12,13,

14,15,16}

{2,6,13, 14,15,16

}

{2,4, 15,16}

{4,6}

{2,6,8}

{6,12}

{6,10 }

{ }

{8, 10, 12}

{6, 12,14}

{6,10,12, 14}TRIAL

1:PLACE 10TH QUEEN IN(10,2)

TRIAL 1 FAILS IN THE 10TH POSITION SO, I TAKE TRIAL 2

OH! NO IT FAILS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 1ST QUEEN IN (1,2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,3,5,6,7,8,9,10,11,12,13

, 14,15,16

}{1,3,4,6,7,8,9,10,11,12,13

, 14,15,16

}{1,3,4,5,7,8,9,10,11,12,13

, 14,15,16

}{1,3,4,5,6,8,9,10,11,12,13

, 14,15,16

}{1,3,4,5,6,7,9,10,11,12,13

, 14,15,16

}

{1,3,4,5,6,7,8,10,11,12,13

, 14,15,16

}

{1,3,4,5,6,7,8,9,11,12,13, 14,15,16

}

{1,3,4,5,6,7,8,9,10,12,13, 14,15,16

}

{1,3,4,5,6,7,8,9,1

0, 11,12,13,15,16}

{1,3,4,5,6,7,8,9,1

0, 11,12,14,15,16}

{1,3,4,5,6,7,8,9,10, 11,12, 14,15,16

}

{1,3,4,5,6,7,8,9,1

0, 11,12,13, 14,16}

{1,3,4,5,6,7,8,9,1

0, 11,12,13, 14,15}

{1,3,4,5,6,7,8,9,1

0, 11,12,13

, 14,15,16

}

TRIAL 2:PLACE 1ST QUEEN IN (1,2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 2ND QUEEN IN (2,4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,3,7,8,

9,10,11,12,13,

14,15,16}

{3,5,8,9, 10,11,12

,13, 14,15,16

}

{1,3,5,6,9,10,11,12,13,

14,15,16}

{1,3,5,6,7,10,11,12,13,

14,15,16}

{1,3,5,6,7,8,11,1

2,13, 14,15,16

}

{1,3,5,6,7,8,9,12,

13, 14,15,16

}

{1,3,5,6,7,8,9,10,

13, 14,15,16

}

{1,3,5,6,7,8,9,10, 11,12,13

,16}

{1,3,5,6,7,8,9,10, 11,12,15

,16}

{1,3,5,6,7,8,9,10, 11,14,15

,16}

{1,3,5,6,7,8,9,10, 11,12,13

, 14}

{1,3,5,6,7,8,9,10, 11,12,13, 14,15}

{1,3,5,6,7,8,9,10, 11,12,13

, 14,15,16

}

TRIAL 2:PLACE 2ND QUEEN IN (2,4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 3RD QUEEN IN (3,6)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5,9, 10,11,12

,13, 14,15,16

}

{1,5,10,11,12,13

, 14,15,16

}{1,3,5,7, 11,12,13

, 14,15,16

}{3,5,7,8,

12,13, 14,15,16

}

{1,3,5,7,8,9,13,

14,15,16}

{1,3,5,7,8,9,10,

14,15,16}

{1,3,5,7,8,9,10,

11,12,13}

{1,3,5,7,8,9,10,

11,12,16}

{1,3,5,7,8,9,10,

11,15,16}

{1,3,5,7,8,9,10,

11,12,13, 14}

{1,3,5,7,8,9,10,

11,12,13, 14,15}

{1,3,5,7,8,9,10,

11,12,13,

14,15,16}

TRIAL 2:PLACE 3RD QUEEN IN (3,6)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 4TH QUEEN IN (4,8)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,11,12,13,

14,15,16}

{1,3,7, 12,13,

14,15,16}{3,5,7,

13, 14,15,16

}

{1,5,7,9, 14,15,16

}

{1,3,5,7,9,10,

15,16}

{1,3,5,7,9,10,

11,12,13}

{1,3,5,7,9,10,

11,12}

{3,5,7,9,10,

11,16}

{1,3,5,7,9,10,

11,12,13, 14}

{1,3,5,7,9,10,

11,12,13, 14,15}

{1,3,5,7,9,10,

11,12,13,

14,15,16}

TRIAL 2:PLACE 4TH QUEEN IN (4,8)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 5TH QUEEN IN (5,10)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7,13,

14,15,16}{3,5,

14,15,16}{1,5,7,9,

15,16}

{1,3,7,9 ,16}

{1,3,5,7,9,11,12,

13}

{1,5,7,9, 11,12}

{3,5,7,9, 11}

{3,5,7,9, 11,12,13

, 14}

{1,3,5,7,9,11,12,

13, 14,15}

{1,3,5,7,9,11,12,

13, 14,15,16

}

TRIAL 2:PLACE 5TH QUEEN IN (5,10)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 6TH QUEEN IN (6,12)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 15,16}{1,5,7,1

6}

{1,3,7,9 }

{1,3,5,7,9,11,13}

{1,5,7,9, 11}

{3,5,9, 11}

{3,5,7,9, 11,13,

14}

{1,5,7,9, 11,13, 14,15}

{1,3,5,7,9, 11,13, 14,15,16

}TRIAL 2:PLACE 6TH QUEEN IN (6,12)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 7TH QUEEN IN (7,14)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{1,3,7,9 }

{1,3,7,9, 11,13}

{1,5,7 ,11}

{3,5,9, 11}

{3,5,9, 11,13}

{1,5,7,9, 11,13,15

}

{1,3,7,9, 11,13,15

,16}TRIAL 2:PLACE 7TH QUEEN IN (7,14)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 8TH QUEEN IN (8,16)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{1,3,7,9 }

{1,3,7,9, 13}

{1,5,7 ,11}

{3,5,9, 11}

{3,5,9, 11,13}

{1,5,7, 11,13,15

}

{1,3,7,9, 11,13,15

}TRIAL 2:PLACE 8TH QUEEN IN (8,16)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 9TH QUEEN IN (9,1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{3,7,9, 13}

{5,7 ,11}

{5,9, 11}

{3,5,9, 11,13}

{5,11,13, 15}

{3,7,9, 11,13,15

}TRIAL 2:PLACE 9TH QUEEN IN (9,1)

I THINK IT’S A RIGHT PATH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 10TH QUEEN IN (10,3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{7,9, 13}

{5,7 ,11}

{5,9, 11}

{5,9, 11,13}

{5,11,13, 15}

{7,11,13, 15}TRIAL

2:PLACE 10TH QUEEN IN (10,3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 11TH QUEEN IN (11,5)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{9, 13}

{7,11}

{5,9, 11}

{9, 11,13}

{11,13, 15}

{7,11,13, 15}TRIAL

2:PLACE 11TH QUEEN IN (11,5)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 12TH QUEEN IN (12,7)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{9, 13}

{7,11}

{5,9, 11}

{11,13}

{11,13, 15}

{13, 15}TRIAL 2:PLACE 12TH QUEEN IN (12,7)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 13TH QUEEN IN (13,9)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{9, 13}

{7,11}

{5,9, 11}

{11,13}

{13, 15}

{13, 15}TRIAL 2:PLACE 13TH QUEEN IN (13,9)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 14TH QUEEN IN (14,11)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{9, 13}

{7,11}

{5,9, 11}

{11,13}

{13, 15}

{15}TRIAL 2:PLACE 14TH QUEEN IN (14,11)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 15TH QUEEN IN (15,13)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{9, 13}

{7,11}

{5,9, 11}

{11,13}

{13, 15}

{15}TRIAL 2:PLACE 15TH QUEEN IN (15,13)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TRIAL 2:PLACE 16TH QUEEN IN (16,15)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

X13

X9

X1

X5

X16

X8

X7

X15

X6

X14

X3

X11

X4

X12

X2

X10

{1,2,3,4,5,6,7,8,9,10,11,1

2,13, 14,15,16

}

{4,5,6,7,8,9,10,11,12,13, 14,15,16

}

{1,6,7,8,9,10,11,12,13,

14,15,16} {1,8,9,

10,11,12,13,

14,15,16}

{3,5, 10,11,12

,13, 14,15,16

}

{1,5,12,13,14,15

,16}

{1,3,7, 14,15,16

}

{3,5, 16}

{1,5,7}

{3,7,9 }

{9, 13}

{7,11}

{5,9, 11}

{11,13}

{13, 15}

{15}TRIAL 2:PLACE 16TH QUEEN IN (16,15)

TRIAL 2 IS THE RIGHT SOLUTION

I AM HAPP

Y

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

SOLUTION

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