/*

Crossfigure problem (CSPLib #21) in Picat.

CSPLib problem 21
http://www.csplib.org/Problems/prob021
"""
Crossfigures are the numerical equivalent of crosswords. You have a grid and some
clues with numerical answers to place on this grid. Clues come in several different
forms (for example: Across 1. 25 across times two, 2. five dozen, 5. a square number,
10. prime, 14. 29 across times 21 down ...).
"""

Also, see
http://en.wikipedia.org/wiki/Cross-figure

William Y. Sit: "On Crossnumber Puzzles and The Lucas-Bonaccio Farm 1998
http://scisun.sci.ccny.cuny.edu/~wyscc/CrossNumber.pdf

Bill Williams: Crossnumber Puzzle, The Little Pigley Farm

This model was inspired by the ECLiPSe model written by Warwick Harvey:
http://www.csplib.org/Problems/prob021/models

Data from
http://thinks.com/crosswords/xfig.htm.

This problem is 001 from http://thinks.com/crosswords/xfig.htm
("X" is the blackbox and is fixed to the value of 0)

1  2  3  4  5  6  7  8  9
---------------------------
1  2  _  3  X  4  _  5  6    1
7  _  X  8  _  _  X  9  _    2
_  X  10 _  X  11 12 X  _    3
13 14 _  _  X  15 _  16 _    4
X  _  X  X  X  X  X  _  X    5
17 _  18 19 X  20 21 _ 22    6
_  X  23 _  X  24 _  X  _    7
25 26 X  27 _  _  X  28 _    8
29 _  _  _  X  30 _  _  _    9

1608 9183
60 201 42
3 72 14 1
5360 2866
3     4
4556 1156
9 67 16 8
68 804 48
1008 7332

Model created by Hakan Kjellerstrand, hakank@gmail.com

*/

% Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/

import util.
import cp.

main => go.

go =>

N = 9,

% Domain = 0..9999, % the max length of the numbers in this problem is 4

% The valid squares (or rather the invalid are marked as "x")
Valid =
[[ _,  _,  _,  _,  x, _, _, _, _],
[ _,  _,  x,  _,  _, _, x, _, _],
[ _,  x,  _,  _,  x, _, _, x, _],
[ _,  _,  _,  _,  x, _, _, _, _],
[ x,  _,  x,  x,  x, x, x, _, x],
[ _,  _,  _,  _,  x, _, _, _, _],
[ _,  x,  -,  _,  x, _, _, x, _],
[ _,  _,  x,  _,  _, _, x, _, _],
[ _,  _,  _,  _,  x, _, _, _, _]],

M = new_array(N,N),
MVars = array_matrix_to_list(M),
MVars :: 0..9,

AList = [A1,A4,A7,A8,A9,A10,A11,A13,A15,A17,A20,A23,A24,A25,A27,A28,A29,A30],
AList :: 0..9999,
DList = [D1,D2,D3,D4,D5,D6,D10,D12,D14,D17,D18,D19,D20,D21,D22,D26,D28],
DList :: 0..9999,

% Set up the constraints between the matrix elements and the
% clue numbers.
across(M, A1, 4, 1, 1),
across(M, A4, 4, 1, 6),
across(M, A7, 2, 2, 1),
across(M, A8, 3, 2, 4),
across(M, A9, 2, 2, 8),
across(M, A10, 2, 3, 3),
across(M, A11, 2, 3, 6),
across(M, A13, 4, 4, 1),
across(M, A15, 4, 4, 6),
across(M, A17, 4, 6, 1),
across(M, A20, 4, 6, 6),
across(M, A23, 2, 7, 3),
across(M, A24, 2, 7, 6),
across(M, A25, 2, 8, 1),
across(M, A27, 3, 8, 4),
across(M, A28, 2, 8, 8),
across(M, A29, 4, 9, 1),
across(M, A30, 4, 9, 6),

down(M, D1, 4, 1, 1),
down(M, D2, 2, 1, 2),
down(M, D3, 4, 1, 4),
down(M, D4, 4, 1, 6),
down(M, D5, 2, 1, 8),
down(M, D6, 4, 1, 9),
down(M, D10, 2, 3, 3),
down(M, D12, 2, 3, 7),
down(M, D14, 3, 4, 2),
down(M, D16, 3, 4, 8),
down(M, D17, 4, 6, 1),
down(M, D18, 2, 6, 3),
down(M, D19, 4, 6, 4),
down(M, D20, 4, 6, 6),
down(M, D21, 2, 6, 7),
down(M, D22, 4, 6, 9),
down(M, D26, 2, 8, 2),
down(M, D28, 2, 8, 8),

% Set up the clue constraints.
%  Across
%  1 27 across times two
%  4 4 down plus seventy-one
%  7 18 down plus four
%  8 6 down divided by sixteen
%  9 2 down minus eighteen
% 10 Dozen in six gross
% 11 5 down minus seventy
% 13 26 down times 23 across
% 15 6 down minus 350
% 17 25 across times 23 across
% 20 A square number
% 23 A prime number
% 24 A square number
% 25 20 across divided by seventeen
% 27 6 down divided by four
% 28 Four dozen
% 29 Seven gross
% 30 22 down plus 450

A1 #= 2 * A27,
A4 #= D4 + 71,
A7 #= D18 + 4,
A8 #= D6 // 16,
A9 #= D2 - 18,
A10 #= 6 * 144 // 12,
A11 #= D5 - 70,
A13 #= D26 * A23,
A15 #= D6 - 350,
A17 #= A25 * A23,

square(A20),
is_prime(A23),
square(A24),
A25 #= A20 // 17,
A27 #= D6 // 4,
A28 #= 4 * 12,
A29 #= 7 * 144,
A30 #= D22 + 450,

% Down
%
%  1 1 across plus twenty-seven
%  2 Five dozen
%  3 30 across plus 888
%  4 Two times 17 across
%  5 29 across divided by twelve
%  6 28 across times 23 across
% 10 10 across plus four
% 12 Three times 24 across
% 14 13 across divided by sixteen
% 16 28 down times fifteen
% 17 13 across minus 399
% 18 29 across divided by eighteen
% 19 22 down minus ninety-four
% 20 20 across minus nine
% 21 25 across minus fifty-two
% 22 20 down times six
% 26 Five times 24 across
% 28 21 down plus twenty-seven

D1 #= A1 + 27,

D2 #= 5 * 12,

D3 #= A30 + 888,
D4 #= 2 * A17,

D5 #= A29 // 12,
D6 #= A28 * A23,
D10 #= A10 + 4,

D12 #= A24 * 3,
D14 #= A13 // 16,
D16 #= 15 * D28,
D17 #= A13 - 399,
D18 #= A29 // 18,
D19 #= D22 - 94,
D20 #= A20 - 9,
D21 #= A25 - 52,
D22 #= 6 * D20,
D26 #= 5 * A24,
D28 #= D21 + 27,

% Fix the black boxes
foreach(I  in 1..N)
foreach(J in 1..N)
if Valid[I,J] == x then
M[I,J] #= 0
end
end
end,

Vars = MVars ++ AList ++ DList,
solve(Vars),

foreach(I in 1..N)
foreach(J in 1..N)
if Valid[I,J] == x then
printf(" ")
else
printf("%d", M[I,J])
end
end,
nl
end,
nl.

/*
across(Matrix, Across, Len, Row, Col)
Constrains 'Across' to be equal to the number represented by the
'Len' digits starting at position (Row, Col) of the array 'Matrix'
and proceeding across.
*/
across(Matrix, Across, Len, Row, Col) =>
Tmp = new_list(Len),
Tmp :: 0..9999,
to_num(Tmp, Across),
foreach(I in 0..Len-1)
Matrix[Row,Col+I] #= Tmp[I+1]
end.

/*
down(Matrix, Down, Len, Row, Col):
Constrains 'Down' to be equal to the number represented by the
'Len' digits starting at position (Row, Col) of the array 'Matrix'
and proceeding down.
*/
down(Matrix, Down, Len, Row, Col) =>
Tmp = new_list(Len),
Tmp :: 0..9999,
to_num(Tmp, Down),
foreach(I in 0..Len-1)
Matrix[Row+I,Col] #= Tmp[I+1]
end.

/*
x is a prime
*/
is_prime(X) =>
Max = fd_max(X),
foreach(I in 2..Max // 2)
I #!= X #=> X mod I #> 0
end.

%
% x is a square
%
square(X) =>
Max = fd_max(X),
Tmp :: 0..Max,
X #= Tmp**2.

to_num(List, Num) =>
to_num(List, 10, Num).

to_num(List, Base, Num) =>
Len = length(List),
Num #= sum([List[I]*Base**(Len-I) : I in 1..Len]).