This should probably be found in
Program above, with comments, as posted on STack Overflow
What happens is:
- sudoku/1 imposes constraints on a 9x9 matrix of elements
- with the help of blocks/2
- problem/2 lists a particular problem
And we can then put both together:
?- problem(1,Matrix), % Matrix is now a 9x9 matrix of partially bound elements. % Impose constraints; these suffice to find a unique solution sudoku(Matrix). Matrix = [[9,8,7,6,5,4,3,2,1], [2,4,6,1,7,3,9,8,5], [3,5,1,9,2,8,7,4,6], [1,2,8,5,3,7,6,9,4], [6,3,4,8,9,2,1,5,7], [7,9,5,4,6,1,8,3,2], [5,1,9,2,8,6,4,7,3], [4,7,2,3,1,9,5,6,8], [8,6,3,7,4,5,2,1,9]].
The constraint solver starts to work immediately (apparently it detects that there is enough information to proceed; is that always the case?). There is no need to call label/1.
Let's see how the constraints between the matrix elements are set up:
sudoku(Rows) :- % If "Rows" is unbound, bind it to a list of 9 unbound variables. % If "Rows" is bound (as in our call above), this just verifies that there % are indeed 9 elements in Rows. Same as: % % Rows = [R1, % R2, % R3, % R4, % R5, % R6, % R7, % R8, % R9]. length(Rows, 9), % For each element in "Rows", bind the element to a list of the same % length as "Rows" (i.e. 9); we now have a 9 x 9 matrix of possibly % unbound variables. A bit too clever maybe. % % Same as: % % Rows = [[E11,E12,E13,E14,E15,E16,E17,E18,E19], % [E21,E22,E23,E24,E25,E26,E27,E28,E29], % [E31,E32,E33,E34,E35,E36,E37,E38,E39], % [E41,E42,E43,E44,E45,E46,E47,E48,E49], % [E51,E52,E53,E54,E55,E56,E57,E58,E59], % [E61,E62,E63,E64,E65,E66,E67,E68,E69], % [E71,E72,E73,E74,E75,E76,E77,E78,E79], % [E81,E82,E83,E84,E85,E86,E87,E88,E89], % [E91,E92,E93,E94,E95,E96,E97,E98,E99]]. maplist(same_length(Rows), Rows), % Concatenate all the variables in all the rows into a new list Vs (that % predicate is badly named, it concatenates lists form a list into a list) append(Rows, Vs), % A first constraint: every element Exx in Vs must be in domain [1..9] % If we passed in a matrix Rows with concrete values for Exx that are not % in domain [1..9], this will fail. Vs ins 1..9, % Another constraint: For every row (list of 9 elements) in Rows: % all the elements of the row must be distinct maplist(all_distinct, Rows), % We now want to impose the constraint that for every column, % all the elements of the column must be distinct % Transpose the 9x9 matrix Rows into the 9x9 matrix Columns. % (the new rows of Columns are the columns of Rows). % % Columns = [[E11,E21,E31,E41,E51,E61,E71,E81,E91], % [E12,E22,E32,E42,E52,E62,E72,E82,E92] % [E13,E23,E33,E43,E53,E63,E73,E83,E93] % [E14,E24,E34,E44,E54,E64,E74,E84,E94] % [E15,E25,E35,E45,E55,E65,E75,E85,E95] % [E16,E26,E36,E46,E56,E66,E76,E86,E96] % [E17,E27,E37,E47,E57,E67,E77,E87,E97] % [E18,E28,E38,E48,E58,E68,E78,E88,E98] % [E19,E29,E39,E49,E59,E69,E79,E89,E99]]. transpose(Rows, Columns), % Another constraint: For every "column of Rows" % all the elements of the column must be distinct maplist(all_distinct, Columns), % Now we need to impose the "all-distinct" constraint % on the 3x3 sub-matrices % Give the rows distinct names Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is], % Impose constraint on "3-element-wide" groups of the % "3-row-high" group of As, Bs, Cs: namely, "all distinct" blocks(As, Bs, Cs), % Same of the next "3-high" group blocks(Ds, Es, Fs), % Same of the next "3-high" group blocks(Gs, Hs, Is).