16-Queen's Problem
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![Page 1: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/1.jpg)
ASSIGNMENT-ARTIFICIAL
INTELLIGENCE
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16-QUEENS PROBLEM USING CSP
K.INIYACSE PREFINAL YEAR
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The Queen can move any number of vacant squares diagonally, horizontally and vertically.
RULES:
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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.,
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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
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CONSTRAINTS
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{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
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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
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{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)
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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
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{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)
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TRIAL 2:PLACE 3RD QUEEN IN (3,6)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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{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)
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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
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{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)
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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
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{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)
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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
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{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)
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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
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{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)
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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
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{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)
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TRIAL 2:PLACE 9TH QUEEN IN (9,1)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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{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
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TRIAL 2:PLACE 10TH QUEEN IN (10,3)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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{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)
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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
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X1
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X8
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X11
X4
X12
X2
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{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)
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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
![Page 31: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/31.jpg)
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)
![Page 32: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/32.jpg)
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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
![Page 33: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/33.jpg)
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)
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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
![Page 35: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/35.jpg)
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)
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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
![Page 37: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/37.jpg)
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)
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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
![Page 39: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/39.jpg)
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
![Page 40: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/40.jpg)
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
SOLUTION
![Page 41: 16-Queen's Problem](https://reader033.fdocuments.net/reader033/viewer/2022061219/54b984c64a79592d038b4597/html5/thumbnails/41.jpg)