16 queens problem - trial 2
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Transcript of 16 queens problem - trial 2
16 QUEENS PROBLEM USING CSP – II K. INIYA
III CSESRI VIDYA COLLEGE OF ENGINEERING AND
TECHNOLOGY, VIRUDHUNAGAR
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
X7
X15
X6
X14
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