N-Queens

The N-Queens is a famous constraint solving benchmark puzzle. It is a generalisation of the original eight queens puzzle, where the goal is to place eight queens on a 8*8 chessboard so that no two queens attach each other.

Here is one way to encode the N-Queens puzzle in B.

MACHINE NQueens
CONSTANTS n,queens
PROPERTIES
 n = 40 & 
 queens : 1..n >-> 1..n /* for each column the row in which the queen is in */
 &
 !(q1,q2).(q1:1..n & q2:2..n & q2>q1
    => queens(q1)+q2-q1 /= queens(q2) & queens(q1)-q2+q1 /= queens(q2))
END

ProB 1.3.7 can solve this puzzle in about 0.150 seconds (for n=40, on a 1.7 GHz Mac Book Air).

One can use graphical visualisation features to display the solution, by declaring the ANIMATION FUNCTION as follows:

MACHINE NQueens40
CONSTANTS n,queens
PROPERTIES
 n = 40 & 
 queens : 1..n >-> 1..n /* for each column the row in which the queen is in */
 &
 !(q1,q2).(q1:1..n & q2:2..n & q2>q1
    => queens(q1)+q2-q1 /= queens(q2) & queens(q1)-q2+q1 /= queens(q2))
DEFINITIONS
      ANIMATION_FUNCTION_DEFAULT == ( {r,c,i|r:1..n & c:1..n & i=(r+c) mod 2 }  );
      ANIMATION_FUNCTION == ( {r,c,i|c:1..n & r=queens(c) & i=2+((r+c) mod 2) }  );
      ANIMATION_IMG0 == "images/sm_empty_box.gif";
      ANIMATION_IMG1 == "images/sm_gray_box.gif";
      ANIMATION_IMG2 == "images/sm_queen_white.gif";
      ANIMATION_IMG3 == "images/sm_queen_black.gif";
      SET_PREF_CLPFD == TRUE;
END

This will lead to ProB to show the solution graphically, as follows (the screenshot is unfortunately cropped and does not show all rows):

ProB Queens 40 Screenshot.png


Instead of using ProB's standard constraint-solving backend, you can also use our Kodkod backend to solve this puzzle. To do this, either "Enable Kodkod for Properties" in the ProB Tcl/Tk "Preferences" menu or add the following to the DEFINITIONS in the machine file above:

    SET_PREF_KODKOD == TRUE

However, using the MiniSat backend solving the N-queens puzzle for n=40 takes 57.1 seconds (on the same hardware as above).