part2.eprime

language ESSENCE' 1.0

$ Takes board size as a parameter.
given BOARD_SIZE : int

letting RANGE be domain int(1..BOARD_SIZE)

$ Results represented as a one-dimensional array with indexes corresponding to columns and content to rows (zero for no queen in that column).
find result : matrix indexed by [RANGE] of int(0..BOARD_SIZE)

$ We want as few queens as possible - as few non-zero column assignments as possible.
minimising sum col : RANGE . (result[col] != 0)

such that

$ Queens do not attack each other. One queen per column by design.
$ Row constraints for the number of queens.
alldifferent_except(result,0),

$ Constraints for / diagonals. Slice the elements whose row index + column index = the same number, as these represent / diagonals.
allDiff([col + result[col] | col : RANGE]),

$ Constraints for \ diagonals.
forAll col : int(1..BOARD_SIZE-1) .
	forAll col2 : int(col+1..BOARD_SIZE) . ((col2 - col) != (result[col2] - result[col])),

$ All fields must be under attack - loop through all and check whether there is a queen on the same column/row/diagonal.
forAll col : RANGE .
	forAll row : RANGE .
		((result[col] != 0) + 
			(exists col2 : RANGE .
				(row = result[col2]) \/ ((result[col2] != 0) /\ (|row - result[col2]| = |col - col2|)))) >= 1