% This is a demo on how easy it is to produce a list of subgrids from a
% Sudoku grid.

% We rely on the library predicate transpose/2, and the predicate chop/3 from
% Homework P5.

% To make it easier to understand the transformation we define a helper
% predicate that generates a matrix filled with two digit numbers RC, where
% R is the row number and C is the column number.

| ?- N = 4, sample_grid(N, G).
N = 4,
G = 
[ [11,12,13,14],
  [21,22,23,24],
  [31,32,33,34],
  [41,42,43,44]
] ? 
yes

% Here is what chop does to a 4 element list:

| ?- chop(2, [a,b,c,d], L).
L = 
[ [a,
   b
],
  [c,
   d
]
] ? 
yes


% Step 1: 

% step1(N, Grid, ChoppedGrid): chop Grid to chunks of size sqrt(N),
% resulting in ChoppedGrid

| ?- N = 4, sample_grid(N, G), step1(N, G, ChoppedG).
N = 4,
G = 
[ [11,12,13,14],
  [21,22,23,24],
  [31,32,33,34],
  [41,42,43,44]
],
ChoppedG = 
[ 
[ [11,12,13,14],
  [21,22,23,24]
],
  
[ [31,32,33,34],
  [41,42,43,44]
]
] ? 
yes


% Step 2: 

% step2(ChoppedG, TChoppedG): transpose each element of list ChoppedG,
% resulting in TChoppedG.

| ?- N = 4, sample_grid(N, G), step1(N, G, ChoppedG), step2(ChoppedG, TChG).
N = 4,
G = 
[ [11,12,13,14],
  [21,22,23,24],
  [31,32,33,34],
  [41,42,43,44]
],
ChoppedG = 
[ 
[ [11,12,13,14],
  [21,22,23,24]
],
  
[ [31,32,33,34],
  [41,42,43,44]
]
],
TChG = 
[ 
[ [11,21],
  [12,22],
  [13,23],
  [14,24]
],
  
[ [31,41],
  [32,42],
  [33,43],
  [34,44]
]
] ? 
yes


% Step 3: 

% step3(N, TChG, SGs): Convert TChG, the output of Step 2, to the list of
% subgrids SGs.

% A possible approach: flatten TChG, then chop it to chunks of size N.

% Note that the flatten/2 predicate of SWI Prolog is not available in
% SICStus. Write your own one, or use append/2.


| ?- N = 4, sample_grid(N, G), step1(N, G, ChG), step2(ChG, TChG), step3(N, TChG, SGs).
N = 4,
G = 
[ [11,12,13,14],
  [21,22,23,24],
  [31,32,33,34],
  [41,42,43,44]
],
ChG = 
[ 
[ [11,12,13,14],
  [21,22,23,24]
],
  
[ [31,32,33,34],
  [41,42,43,44]
]
],
TChG = 
[ 
[ [11,21],
  [12,22],
  [13,23],
  [14,24]
],
  
[ [31,41],
  [32,42],
  [33,43],
  [34,44]
]
],
SGs = 
[ [11,21,12,22],
  [13,23,14,24],
  [31,41,32,42],
  [33,43,34,44]
] ? 
yes

% Example 2: obtaining the subgrids of a 9 x 9 Sudoku grid.


| ?- N = 9, sample_grid(N, G), step1(N, G, ChG), step2(ChG, TChG), step3(N, TChG, SGs).
N = 9,
G = 
[ [11,12,13,14,15,16,17,18,19],
  [21,22,23,24,25,26,27,28,29],
  [31,32,33,34,35,36,37,38,39],
  [41,42,43,44,45,46,47,48,49],
  [51,52,53,54,55,56,57,58,59],
  [61,62,63,64,65,66,67,68,69],
  [71,72,73,74,75,76,77,78,79],
  [81,82,83,84,85,86,87,88,89],
  [91,92,93,94,95,96,97,98,99]
],
ChG = 
[ 
[ [11,12,13,14,15,16,17,18,19],
  [21,22,23,24,25,26,27,28,29],
  [31,32,33,34,35,36,37,38,39]
],
  
[ [41,42,43,44,45,46,47,48,49],
  [51,52,53,54,55,56,57,58,59],
  [61,62,63,64,65,66,67,68,69]
],
  
[ [71,72,73,74,75,76,77,78,79],
  [81,82,83,84,85,86,87,88,89],
  [91,92,93,94,95,96,97,98,99]
]
],
TChG = 
[ 
[ [11,21,31],
  [12,22,32],
  [13,23,33],
  [14,24,34],
  [15,25,35],
  [16,26,36],
  [17,27,37],
  [18,28,38],
  [19,29,39]
],
  
[ [41,51,61],
  [42,52,62],
  [43,53,63],
  [44,54,64],
  [45,55,65],
  [46,56,66],
  [47,57,67],
  [48,58,68],
  [49,59,69]
],
  
[ [71,81,91],
  [72,82,92],
  [73,83,93],
  [74,84,94],
  [75,85,95],
  [76,86,96],
  [77,87,97],
  [78,88,98],
  [79,89,99]
]
],
SGs = 
[ [11,21,31,12,22,32,13,23,33],
  [14,24,34,15,25,35,16,26,36],
  [17,27,37,18,28,38,19,29,39],
  [41,51,61,42,52,62,43,53,63],
  [44,54,64,45,55,65,46,56,66],
  [47,57,67,48,58,68,49,59,69],
  [71,81,91,72,82,92,73,83,93],
  [74,84,94,75,85,95,76,86,96],
  [77,87,97,78,88,98,79,89,99]
] ? 
yes
| ?-