Board Assembly with constraints

Question

I am doing this problem but I am completely new to Prolog and I have no idea how to do it.

Nine parts of an electronic board have square shape, the same size and each edge of every part is marked with a letter and a plus or minus sign. The parts are to be assembled into a complete board as shown in the figure below such that the common edges have the same letter and opposite signs. Write a planner in Prolog such that the program takes 'assemble' as the query and outputs how to assemble the parts, i.e. determine the locations and positions of the parts w.r.t. the current positions so that they fit together to make the complete board.

I have tried solving it and I have written the following clauses:

complement(a,aNeg).
complement(b,bNeg).
complement(c,cNeg).
complement(d,dNeg).
complement(aNeg,a).
complement(bNeg,b).
complement(cNeg,c).
complement(dNeg,d).
% Configuration of boards, (board,left,top,right,bottom)
conf(b1,aNeg,bNeg,c,d).
conf(b2,bNeg,a,d,cNeg).
conf(b3,dNeg,cNeg,b,d).
conf(b4,b,dNeg,cNeg,d).
conf(b5,d,b,cNeg,aNeg).
conf(b6,b,aNeg,dNeg,c).
conf(b7,aNeg,bNeg,c,b).
conf(b8,b,aNeg,cNeg,a).
conf(b9,cNeg,bNeg,a,d).

position(b1,J,A).
position(b2,K,B).
position(b3,L,C).
position(b4,M,D).
position(b5,N,E).
position(b6,O,F).
position(b7,P,G).
position(b8,Q,H).
position(b9,R,I).

assemble([A,B,C,E,D,F,G,H,I,J,K,L,M,N,O,P,Q,R]) :- 
    Variables=[(A,J),(B,K),(C,L),(D,M),(E,N),(F,O),(G,P),(H,Q),(I,R)],
    all_different(Variables),
    A in 1..3, B in 1..3, C in 1..3, D in 1..3, E in 1..3,
    F in 1..3, G in 1..3, H in 1..3, I in 1..3, J in 1..3,
    K in 1..3, L in 1..3, M in 1..3, N in 1..3, O in 1..3,
    P in 1..3, Q in 1..3, R in 1..3,
    % this is where I am stuck, what to write next

I don't know even if they are correct and I am not sure how to proceed further to solve this problem.

Answer 1

Trivial with CLP(FD):

:- use_module(library(clpfd)).

board(Board) :-
    Board = [[A1,A2,A3],
             [B1,B2,B3],
             [C1,C2,C3]],
    maplist(top_bottom, [A1,A2,A3], [B1,B2,B3]),
    maplist(top_bottom, [B1,B2,B3], [C1,C2,C3]),
    maplist(left_right, [A1,B1,C1], [A2,B2,C2]),
    maplist(left_right, [A2,B2,C2], [A3,B3,C3]),
    pieces(Ps),
    maplist(board_piece(Board), Ps).

top_bottom([_,_,X,_], [Y,_,_,_]) :- X #= -Y.

left_right([_,X,_,_], [_,_,_,Y]) :- X #= -Y.

pieces(Ps) :-
    Ps = [[-2,3,4,-1], [1,4,-3,-4], [-3,2,4,-4],
          [-4,-3,4,2], [2,-3,-1,4], [-1,-4,3,2],
          [-2,3,2,-1], [-1,-3,1,2], [-2,1,4,-3]].

board_piece(Board, Piece) :-
    member(Row, Board),
    member(Piece0, Row),
    rotation(Piece0, Piece).

rotation([A,B,C,D], [A,B,C,D]).
rotation([A,B,C,D], [B,C,D,A]).
rotation([A,B,C,D], [C,D,A,B]).
rotation([A,B,C,D], [D,A,B,C]).

Example query and its result:

?- time(board(Bs)), maplist(writeln, Bs).
11,728,757 inferences, 0.817 CPU in 0.817 seconds
[[-3, -4, 1, 4], [-1, -2, 3, 4], [4, -4, -3, 2]]
[[-1, 4, 2, -3], [-3, 4, 2, -4], [3, 2, -1, -4]]
[[-2, 1, 4, -3], [-2, 3, 2, -1], [1, 2, -1, -3]]

This representation uses 1,2,3,4 to denote positive a,b,c,d, and -1,-2,-3,-4 for the negative ones.