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The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EibkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGNS9GOVEnbm9ybWFsRic=-Queens problem is to place n queens on an n by n chessboard such that no two queens are mutually attacking (are in the same row, column, or diagonal). The above image shows the squares attacked by a queen on an 8 by 8 chessboard. The problem is solvable for all n larger than 3. We can use Maple's built-in efficient SAT solver to quickly solve this problem. A SAT solver takes as input a formula in Boolean logic and returns an assignment to the variables which makes the formula true (if one exists). See the Satisfy command for more information.

We'll use the Boolean variables 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Y1L0Y5USdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EieUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Y1L0Y5USdub3JtYWxGJw== between 1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EibkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Y1L0Y5USdub3JtYWxGJw== to denote if there is a queen on the square at 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. We'll also use a few simple functions to compute the set of variables which are attacked by a queen at LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYoLUkjbWlHRiQ2J1EieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYvUSIsRidGMkY4L0Y8USdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNidRInlGJ0YyRjVGOEY7RjJGOEZCRjJGOEZC. To define 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 we'll use the following four functions: * varsInCol: Returns the set of variables in the same column as the square at LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYoLUkjbWlHRiQ2J1EieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYvUSIsRidGMkY4L0Y8USdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNidRInlGJ0YyRjVGOEY7RjJGOEZCRjJGOEZC. * varsInRow: Returns the set of variables in the same row as the square at LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYoLUkjbWlHRiQ2J1EieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYvUSIsRidGMkY4L0Y8USdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNidRInlGJ0YyRjVGOEY7RjJGOEZCRjJGOEZC. * varsInDiag: Returns the set of variables in the same diagonal as the square at 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. * varsInAntiDiag: Returns the set of variables in the same anti-diagonal as the square at LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYoLUkjbWlHRiQ2J1EieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYvUSIsRidGMkY4L0Y8USdub3JtYWxGJy8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNidRInlGJ0YyRjVGOEY7RjJGOEZCRjJGOEZC.

varsInCol := proc(x, y) return {seq(Q[x, i], i=1..n)}; end proc: varsInRow := proc(x, y) return {seq(Q[i, y], i=1..n)}; end proc: varsInDiag := proc(x, y) local diagVars; if y > n - x then diagVars := {seq(Q[i, x+y-i], i=x+y-n..n)}; else diagVars := {seq(Q[i, x+y-i], i=1..x+y-1)}; end if; return diagVars; end proc: varsInAntiDiag := proc(x, y) local diagVars; if x < y then diagVars := {seq(Q[i, y-x+i], i=1..n-y+x)}; else diagVars := {seq(Q[i, y-x+i], i=x-y+1..n)}; end if; return diagVars; end proc: attackedVars := proc(x, y) return (varsInRow(x, y) union varsInCol(x, y) union varsInDiag(x, y) union varsInAntiDiag(x, y)) minus {Q[x, y]}; end proc:

JSFH

For a given LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EibkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Y1L0Y5USdub3JtYWxGJw== we need to generate the constraints for the n by n chessboard. There are two types of constraints: * Postive constraints (saying there are at least n queens on the board) * Negative constraints (saying each queen doesn't attack any other queen)

Each row must contain a queen since there are n rows and only one queen can go in each row. These clauses have the form 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EiakYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGNS9GOVEnbm9ybWFsRic= from 1 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EibkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGNS9GOVEnbm9ybWFsRic=. Similarly, each column must contain a queen. positiveConstraints := proc(n) local x, constraints; constraints := Array(1..2*n); for x from 1 to n do constraints[2*x-1] := &or(seq(Q[i, x], i=1..n)); constraints[2*x] := &or(seq(Q[x, j], j=1..n)) end do: return &and(entries(constraints, nolist)); end proc:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

If a square contains a queen then no other square in that row, column, or diagonal contains a queen. These clauses have the form 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0Y1L0Y5USdub3JtYWxGJw== is in 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. Equivalently, they have the form 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 where 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 negativeConstraints := proc(n) local x, y, constraints; constraints := Array(1..n, 1..n); for x from 1 to n do for y from 1 to n do constraints[x, y] := &implies(Q[x, y], &not(&or(attackedVars(x, y)[]))); end do; end do; return &and(entries(constraints, nolist)); end proc: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

We use the Satisfy command from the Logic package which finds a satisfying assignment of a logical formula if one exists. We use the Usage command from the CodeTools package to mesaure how quickly the solution is found. n := 8: with(Logic): allConstraints := &and(positiveConstraints(n), negativeConstraints(n)): satisfyingAssignment := CodeTools:-Usage(Satisfy(allConstraints)):

JSFH

The following functions use commands from the plottools package to draw a graphical representation of a chessboard and a chess queen on square 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. with(plottools): drawQueen := proc(x, y) return ellipse([x+.5, y+.35], .24, 0.05, filled = true), rectangle([x+.25, y+.35], [x+.75, y+.2]), ellipse([x+.5, y+.2], .25, 0.05, filled = true), disk([x+.2, y+.7], 0.03), disk([x+.35, y+.725], 0.03), disk([x+.5, y+.75], 0.03), disk([x+.65, y+.725], 0.03), disk([x+.8, y+.7], 0.03), polygon([[.25+x, .25+y], [.2+x, .7+y], [.325+x, .25+y], [.35+x, .725+y], [.4375+x, .25+y], [x+.5, .75+y], [.5625+x, .25+y], [.65+x, .725+y], [.675+x, .25+y], [.8+x, .7+y], [.75+x, .25+y]]) end proc: drawChessboard := proc() return seq(seq(rectangle([i,j+1], [i+1,j], color=`if`(type(i+j, 'even'), gray, white)), j=1..n), i=1..n); end proc: The following code draws the chessboard and the queens on the board in the locations that correspond to the squares which are assigned to true in the satisfying solution that was found. queens := Array(1..n): i := 1: if satisfyingAssignment <> NULL then for eq in satisfyingAssignment do if rhs(eq) then queens[i] := drawQueen(op(lhs(eq))); i := i + 1; end if; end do: plots:-display(entries(queens, nolist), drawChessboard(), scaling=constrained, axes=none); else printf("No solution when n is %d.\134n", n); end if;

JSFHJSFH