1. PROLOG
8 QUEENS PROBLEM

2. 8 QUEENS Place 8 queens on an 8 X 8 chessboard so that:
Queens do not attack each other

3. 8 QUEENS Solution will be programmed as a unary predicate
solution(Position)
Will be true if Position represents a valid chessboard position for the 8 queens and no 2 queens can attack each other

4. 8 QUEENS Representation of the chess board position
Each queen will be represented by X and Y (row and column)
We will define a new operator, :
X:Y
Potential solution:
[X1:Y1, X2:Y2, X3:Y3, …, X8:Y8]

5. 8 QUEENS Since every queen will have to be in a different row 
We can hard code all the Xis to be 1 to 8
A potential solution can be templated as :
[1:Y1, 2:Y2, 3:Y3, .., 8:Y8]

6. 8 QUEENS Recursive formulation of the problem
We will “think” about this problem in terms of placing N queens not just 8
 recursive relation between N queens and (N – 1) queens
Use this to solve when number of queens is 8

7. 8 QUEENS Recursive formulation of the problem
List of N queens
Base case: list of queens is empty
General case: list of queens looks like [X:Y | Others] and
Others itself is a solution
X and Y must be integers between 1 and 8
The X:Y queen must not attack any queen in Others

8. 8 QUEENS solution([]). % no queen  good solution([X:Y | Others]) :-
solution(Others),
member(Y,[1,2,3,4,5,6,7,8]),
noattack(X:Y,Others). % need to define
% X will be hard coded 1 to 8

9. 8 QUEENS Need to define predicate noattack noattack takes 2 arguments:
A queen
A list of queens
Base case: list of queens is empty
The queen cannot attack the queens in the empty list (there is not any)

10. 8 QUEENS Base case for noattack noattack(_,[]).
A queen (1st argument) cannot be attacked by 0 queen

11. 8 QUEENS
General case
If the list of queens is not empty, then it can be written [X1:Y1 | Others] where X1:Y1 represents the first queen (head) and Others the other queens (tail)
noattack(X:Y,[X1:Y1 | Others]) :- ????

12. 8 QUEENS noattack(X:Y,[X1:Y1 | Others]) :- ???? That will be true if:
Queen X:Y cannot attack queen X1:Y1, and
Queen X:Y cannot attack any of the queens in the list of queens Others

13. 8 QUEENS One queen vs another (X:Y vs X1:Y1)
X and X1 must be different but that is automatic (the Xis are hard coded)
Y and Y1 must be different (column)
Y1 – Y must be different from X1 – X (main diagonal)
Y1 – Y must be different from X – X1 (other diagonal)

14. 8 QUEENS One queen vs another (X:Y vs X1:Y1)
Y1 and Y must be different
X1 and X are hard coded and automatically different
Y1 – Y different from X1 – X
Y1 – Y different from X – X1
We will use the =\= operator , arithmetic not equal

15. 8 QUEENS General case for noattack noattack(X:Y,[X1:Y1 | Others]) :-
Y =\= Y1,
Y1 – Y =\= X1 – X,
Y1 – Y =\= X – X1,
noattack(X:Y, Others).
% X and X1 are hard coded different

16. 8 QUEENS solution([]). % no queen solution([X:Y | Others]) :-
solution(Others),
member(Y,[1,2,3,4,5,6,7,8]),
noattack(X:Y,Others).

17. 8 QUEENS noattack(_,[]). noattack(X:Y,[X1:Y1 | Others]) :- Y =\= Y1,
Y1 – Y =\= X1 – X,
Y1 – Y =\= X – X1,
noattack(X:Y, Others).

18. 8 QUEENS Hard code the Xis into 1 to 8
template([1:Y1, 2:Y2, 3:Y3, …, 8:Y8]).
To run the program:
template(S), solution(S).