/* 

  Benjamin Franklin's 8x8 Magic Square in Picat.

  Problem formulation from
  the Formula One model:
  http://www.f1compiler.com/samples/Franklin%27s%208x8%20Magic%20Square.f1.html
  """
  In 2006 in the article published online by the Proceedings of the Royal 
  Society "Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS"
  by Daniel Schindel, Matthew Rempel and Peter Loly
  the authors showed there are 1,105,920 variations of his magic square.
  (http://www.physics.umanitoba.ca/news/loly_paper.pdf)
  """

  Also, see e.g.
  http://www.mathpages.com/home/kmath155.htm


  The following is a simple translation of the F1 model to Picat (via MiniZinc).
  Note: I have not checked that this model generates exactly 
  1,105,920 solutions.


  This Picat model was created by Hakan Kjellerstrand, hakank@gmail.com
  See also my Picat page: http://www.hakank.org/picat/

*/

import cp.


main => go.

go ?=>
    magic_square(A),
    foreach(Row in A) println(Row) end,
    nl,
    fail,

    nl.

go => true.

go2 =>
  Count = count_all(magic_square(_A)),
  println(count=Count),
  nl.

magic_square(A) =>
    N = 8,

    Sum = 260,
    Sum2 = Sum div 2,

    % an array of 64 variables of an 8x8 square  
    A = [[A1, A2, A3, A4, A5, A6, A7, A8],
        [B1, B2, B3, B4, B5, B6, B7, B8],
        [C1, C2, C3, C4, C5, C6, C7, C8],
        [D1, D2, D3, D4, D5, D6, D7, D8],
        [E1, E2, E3, E4, E5, E6, E7, E8],
        [F1, F2, F3, F4, F5, F6, F7, F8],
        [G1, G2, G3, G4, G5, G6, G7, G8],
        [H1, H2, H3, H4, H5, H6, H7, H8]],
    A :: 1..64,

    foreach(I in 1..N)
      all_different([A[I,J] : J in 1..N])
    end,

    foreach(J in 1..N) 
      all_different([A[I,J] : I in 1..N])
    end,

    % constraints to remove symmetries. these have nothing to do 
    % with the Benjamin Franklin's constraints.

    A1 #> A8,  A1 #> H1,  A1 #> H8,  A8 #> H1,       

    % half rows and half columns (32 constraints)

    A1 + A2 + A3 + A4 #= Sum2,   
    B1 + B2 + B3 + B4 #= Sum2,   
    C1 + C2 + C3 + C4 #= Sum2,   
    D1 + D2 + D3 + D4 #= Sum2,   
    A1 + B1 + C1 + D1 #= Sum2,   
    A2 + B2 + C2 + D2 #= Sum2,   
    A3 + B3 + C3 + D3 #= Sum2,   
    A4 + B4 + C4 + D4 #= Sum2,   
    A5 + A6 + A7 + A8 #= Sum2,   
    B5 + B6 + B7 + B8 #= Sum2,   
    C5 + C6 + C7 + C8 #= Sum2,   
    D5 + D6 + D7 + D8 #= Sum2,   
    A5 + B5 + C5 + D5 #= Sum2,   
    A6 + B6 + C6 + D6 #= Sum2,   
    A7 + B7 + C7 + D7 #= Sum2,   
    A8 + B8 + C8 + D8 #= Sum2,   
    E1 + F1 + G1 + H1 #= Sum2,   
    E2 + F2 + G2 + H2 #= Sum2,   
    E3 + F3 + G3 + H3 #= Sum2,   
    E4 + F4 + G4 + H4 #= Sum2,   
    E1 + E2 + E3 + E4 #= Sum2,   
    F1 + F2 + F3 + F4 #= Sum2,   
    G1 + G2 + G3 + G4 #= Sum2,   
    H1 + H2 + H3 + H4 #= Sum2,   
    E5 + E6 + E7 + E8 #= Sum2,   
    F5 + F6 + F7 + F8 #= Sum2,   
    G5 + G6 + G7 + G8 #= Sum2,   
    H5 + H6 + H7 + H8 #= Sum2,   
    E5 + F5 + G5 + H5 #= Sum2,   
    E6 + F6 + G6 + H6 #= Sum2,   
    E7 + F7 + G7 + H7 #= Sum2,   
    E8 + F8 + G8 + H8 #= Sum2,   

    % bent diagonals (8 constraints for each directions)
    %  8 7 6 5 4 3 2 1
    %  7 6 5 4 3 2 1 8
    %  6 5 4 3 2 1 8 7
    %  5 4 3 2 1 8 7 6
    %  5 4 3 2 1 8 7 6
    %  6 5 4 3 2 1 8 7
    %  7 6 5 4 3 2 1 8
    %  1 7 6 5 4 3 2 1
    %  <    
    A8 + B7 + C6 + D5 + E5 + F6 + G7 + H8 #= Sum, 
    A7 + B6 + C5 + D4 + E4 + F5 + G6 + H7 #= Sum, 
    A6 + B5 + C4 + D3 + E3 + F4 + G5 + H6 #= Sum, 
    A5 + B4 + C3 + D2 + E2 + F3 + G4 + H5 #= Sum, 
    A4 + B3 + C2 + D1 + E1 + F2 + G3 + H4 #= Sum, 
    A3 + B2 + C1 + D8 + E8 + F1 + G2 + H3 #= Sum, 
    A2 + B1 + C8 + D7 + E7 + F8 + G1 + H2 #= Sum, 
    A1 + B8 + C7 + D6 + E6 + F7 + G8 + H1 #= Sum, 

    % the four entries in every 2x2 subsquare sum to 130.
    % 8x8 #= 64 constraints

    A1 + A2 + B1 + B2 #= Sum2,  
    A2 + A3 + B2 + B3 #= Sum2,  
    A3 + A4 + B3 + B4 #= Sum2,  
    A4 + A5 + B4 + B5 #= Sum2,  
    A5 + A6 + B5 + B6 #= Sum2,  
    A6 + A7 + B6 + B7 #= Sum2,  
    A7 + A8 + B7 + B8 #= Sum2,  
    A8 + A1 + B8 + B1 #= Sum2,  % wrAp

    B1 + B2 + C1 + C2 #= Sum2,  
    B2 + B3 + C2 + C3 #= Sum2,  
    B3 + B4 + C3 + C4 #= Sum2,  
    B4 + B5 + C4 + C5 #= Sum2,  
    B5 + B6 + C5 + C6 #= Sum2,  
    B6 + B7 + C6 + C7 #= Sum2,  
    B7 + B8 + C7 + C8 #= Sum2,  
    B8 + B1 + C8 + C1 #= Sum2,  % wrAp
    
    C2 + C3 + D2 + D3 #= Sum2,  
    C3 + C4 + D3 + D4 #= Sum2,  
    C4 + C5 + D4 + D5 #= Sum2,  
    C5 + C6 + D5 + D6 #= Sum2,  
    C6 + C7 + D6 + D7 #= Sum2,  
    C7 + C8 + D7 + D8 #= Sum2,  
    C8 + C1 + D8 + D1 #= Sum2,  % wrAp

    D1 + D2 + E1 + E2 #= Sum2,  
    D2 + D3 + E2 + E3 #= Sum2,  
    D3 + D4 + E3 + E4 #= Sum2,  
    D4 + D5 + E4 + E5 #= Sum2,  
    D5 + D6 + E5 + E6 #= Sum2,  
    D6 + D7 + E6 + E7 #= Sum2,  
    D7 + D8 + E7 + E8 #= Sum2,  
    D8 + D1 + E8 + E1 #= Sum2,  % wrAp

    E1 + E2 + F1 + F2 #= Sum2,  
    E2 + E3 + F2 + F3 #= Sum2,  
    E3 + E4 + F3 + F4 #= Sum2,  
    E4 + E5 + F4 + F5 #= Sum2,  
    E5 + E6 + F5 + F6 #= Sum2,  
    E6 + E7 + F6 + F7 #= Sum2,  
    E7 + E8 + F7 + F8 #= Sum2,  
    E8 + E1 + F8 + F1 #= Sum2,  % wrAp

    F1 + F2 + G1 + G2 #= Sum2,  
    F2 + F3 + G2 + G3 #= Sum2,  
    F3 + F4 + G3 + G4 #= Sum2,  
    F4 + F5 + G4 + G5 #= Sum2,  
    F5 + F6 + G5 + G6 #= Sum2,  
    F6 + F7 + G6 + G7 #= Sum2,  
    F7 + F8 + G7 + G8 #= Sum2,  
    F8 + F1 + G8 + G1 #= Sum2,  % wrAp

    G1 + G2 + H1 + H2 #= Sum2,  
    G2 + G3 + H2 + H3 #= Sum2,  
    G3 + G4 + H3 + H4 #= Sum2,  
    G4 + G5 + H4 + H5 #= Sum2,  
    G5 + G6 + H5 + H6 #= Sum2,  
    G6 + G7 + H6 + H7 #= Sum2,  
    G7 + G8 + H7 + H8 #= Sum2,  
    G8 + G1 + H8 + H1 #= Sum2,  % wrAp

    H1 + H2 + A1 + A2 #= Sum2,  
    H2 + H3 + A2 + A3 #= Sum2,  
    H3 + H4 + A3 + A4 #= Sum2,  
    H4 + H5 + A4 + A5 #= Sum2,  
    H5 + H6 + A5 + A6 #= Sum2,  
    H6 + H7 + A6 + A7 #= Sum2,  
    H7 + H8 + A7 + A8 #= Sum2,  
    H8 + H1 + A8 + A1 #= Sum2,  % wrAp

    % bent diagonals (8 constraints for each directions)
    %  1 8 7 6 6 7 8 1
    %  2 1 8 7 7 8 1 2
    %  3 2 1 8 8 1 2 3
    %  4 3 2 1 1 2 3 4
    %  5 4 3 2 2 3 4 5
    %  6 5 4 3 3 4 5 6
    %  7 6 5 4 4 5 6 7
    %  8 7 6 5 5 6 7 8
    %  \/
    A1 + B2 + C3 + D4 + D5 + C6 + B7 + A8 #= Sum, 
    B1 + C2 + D3 + E4 + E5 + D6 + C7 + B8 #= Sum, 
    C1 + D2 + E3 + F4 + F5 + E6 + D7 + C8 #= Sum, 
    D1 + E2 + F3 + G4 + G5 + F6 + E7 + D8 #= Sum, 
    E1 + F2 + G3 + H4 + H5 + G6 + F7 + E8 #= Sum, 
    F1 + G2 + H3 + A4 + A5 + H6 + G7 + F8 #= Sum, 
    G1 + H2 + A3 + B4 + B5 + A6 + H7 + G8 #= Sum,  
    H1 + A2 + B3 + C4 + C5 + B6 + A7 + H8 #= Sum, 

    % bent diagonals (8 constraints for each directions)
    %  8 7 6 5 5 6 7 8
    %  7 6 5 4 4 5 6 7
    %  6 5 4 3 3 4 5 6
    %  5 4 3 2 2 3 4 5
    %  4 3 2 1 1 2 3 4
    %  3 2 1 8 8 1 2 3
    %  2 1 8 7 7 8 1 2
    %  1 8 7 6 6 7 8 1
    % ,
    H1 + G2 + F3 + E4 + E5 + F6 + G7 + H8 #= Sum, 
    G1 + F2 + E3 + D4 + D5 + E6 + F7 + G8 #= Sum, 
    F1 + E2 + D3 + C4 + C5 + D6 + E7 + F8 #= Sum, 
    E1 + D2 + C3 + B4 + B5 + C6 + D7 + E8 #= Sum, 
    D1 + C2 + B3 + A4 + A5 + B6 + C7 + D8 #= Sum, 
    C1 + B2 + A3 + H4 + H5 + A6 + B7 + C8 #= Sum, 
    B1 + A2 + H3 + G4 + G5 + H6 + A7 + B8 #= Sum, 
    A1 + H2 + G3 + F4 + F5 + G6 + H7 + A8 #= Sum, 

    % bent diagonals (8 constraints for each directions)
    %  1 2 3 4 5 6 7 8
    %  8 1 2 3 4 5 6 7
    %  7 8 1 2 3 4 5 6
    %  6 7 8 1 2 3 4 5
    %  6 7 8 1 2 3 4 5
    %  7 8 1 2 3 4 5 6
    %  8 1 2 3 4 5 6 7
    %  1 2 3 4 5 6 7 8
    %  #>
    A1 + B2 + C3 + D4 + E4 + F3 + G2 + H1 #= Sum, 
    A2 + B3 + C4 + D5 + E5 + F4 + G3 + H2 #= Sum, 
    A3 + B4 + C5 + D6 + E6 + F5 + G4 + H3 #= Sum, 
    A4 + B5 + C6 + D7 + E7 + F6 + G5 + H4 #= Sum,    
    A5 + B6 + C7 + D8 + E8 + F7 + G6 + H5 #= Sum,    
    A6 + B7 + C8 + D1 + E1 + F8 + G7 + H6 #= Sum,    
    A7 + B8 + C1 + D2 + E2 + F1 + G8 + H7 #= Sum,    
    A8 + B1 + C2 + D3 + E3 + F2 + G1 + H8 #= Sum,  

    solve(A.vars).