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Just for a reminder, here is the definition of the top level
(defun SkuSolve (partial_solution) (let ((sudoku (SkuInitialize (SkuNew) partial_solution))) (printf "starting with: \n%s\n" (SkuPrint sudoku)) (printf "\nfound solution:\n%s\n" (SkuPrint (SkuFindSolution sudoku)))))
We have already seen the definitions
of SkuNew, SkuInitialize,
and SkuPrint. Now we can take a look at the
SkuFindSolution, the function which actually searches for
the sudoku solution.
The function SkuFindSolution takes an instance of
class SkuSudoku (created by SkuNew, and
populated by SkuInitialize) and modifies it to find a
solution of the sudoku puzzle, assuming one exists.
How does it work?
It basically brute forces its way through the cells in the board,
trying every possibility for each cell, from 1 to 9, which does not
present an immediate conflict. If no solution exists for a
particular cell (i.e., if each choice from 1 to 9 inflicts a
conflict), the algorithm backtracks and tries a different guess, until
it guesses correctly.
The local function conflict? asks "Does the given digit
already exist in the row, column, or 3x3 block which contains the
cell?" If not, it is a potentially valid guess. The local
function solve_cells takes a list of all the remaining
cells which have not yet been visited. Each digit that does not create a conflict is tried. There are three cases in the (cond
(defun SkuFindSolution (partial) (prog () (labels ((conflict? (digit cell) (exists group '(column row b3x3) (exists c (slotValue cell group)->cells (and (neq c cell) (eqv digit c->value))))) (solve_cells (cells) (cond ((null cells) (return sudoku)) (((car cells)->value) (and (not (conflict? (car cells)->value (car cells))) (solve_cells (cdr cells)))) (t (let ((cell (car cells))) (for solution 1 9 (unless (conflict? solution cell) cell->value = solution (solve_cells (cdr cells)))) cell->value = nil))))) (solve_cells sudoku->cells))))
Now you can test the program by copying the following into the CIWindow which
comes from the Sudoku
(SkuSolve '((5 3 ? ? 7 ? ? ? ?) (6 ? ? 1 9 5 ? ? ?) (? 9 8 ? ? ? ? 6 ?) (8 ? ? ? 6 ? ? ? 3) (4 ? ? 8 ? 3 ? ? 1) (7 ? ? ? 2 ? ? ? 6) (? 6 ? ? ? ? 2 8 ?) (? ? ? 4 1 9 ? ? 5) (? ? ? ? 8 ? ? 7 9)))
You should get the following result if you entered the code correctly.
starting with: +-----------------+|5|3| | |7| | | | ||6| | |1|9|5| | | || |9|8| | | | |6| ||8| | | |6| | | |3||4| | |8| |3| | |1||7| | | |2| | | |6|| |6| | | | |2|8| || | | |4|1|9| | |5|| | | | |8| | |7|9|+-----------------+found solution:+-----------------+|5|3|4|6|7|8|9|1|2||6|7|2|1|9|5|3|4|8||1|9|8|3|4|2|5|6|7||8|5|9|7|6|1|4|2|3||4|2|6|8|5|3|7|9|1||7|1|3|9|2|4|8|5|6||9|6|1|5|3|7|2|8|4||2|8|7|4|1|9|6|3|5||3|4|5|2|8|6|1|7|9|+-----------------+
If you notice, this algorithm only finds one solution of the sudoku
puzzle. In some cases there are multiple solutions, but this
algorithm won't find them. For example, the empty puzzle has lots of
(SkuSolve '((? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?) (? ? ? ? ? ? ? ? ?)))
starting with: +-----------------+| | | | | | | | | || | | | | | | | | || | | | | | | | | || | | | | | | | | || | | | | | | | | || | | | | | | | | || | | | | | | | | || | | | | | | | | || | | | | | | | | |+-----------------+found solution:+-----------------+|9|7|8|5|3|1|6|4|2||6|4|2|9|7|8|5|3|1||5|3|1|6|4|2|9|7|8||8|9|7|2|1|4|3|6|5||3|6|5|8|9|7|2|1|4||2|1|4|3|6|5|8|9|7||7|8|9|1|2|3|4|5|6||4|5|6|7|8|9|1|2|3||1|2|3|4|5|6|7|8|9|+-----------------+
The SkuSolve function is good at detecting that no
solution exists for a particular puzzle.
(SkuSolve '((5 3 4 6 7 8 9 1 2) (6 7 2 1 9 5 3 4 8) (1 9 8 3 4 2 5 6 7) (8 5 9 7 6 1 4 2 3) (4 2 6 8 5 3 7 9 1) (7 1 3 9 2 4 8 5 6) (9 6 ? 5 3 7 2 8 4) (1 8 7 4 1 9 6 3 5) (? ? 5 2 8 6 1 7 9)))
starting with: +-----+-----+-----+|5|3|4|6|7|8|9|1|2||6|7|2|1|9|5|3|4|8||1|9|8|3|4|2|5|6|7||8|5|9|7|6|1|4|2|3||4|2|6|8|5|3|7|9|1||7|1|3|9|2|4|8|5|6||9|6| |5|3|7|2|8|4||1|8|7|4|1|9|6|3|5|| | |5|2|8|6|1|7|9|+-----+-----+-----+no solution
In this posting, we have seen a fairly straightforward approach to
solving the sudoku puzzle. The solution algorithm is pretty easy
because the data structures representing the structure of the board
make it easy to ask the questions we need to ask, such as:
The particular implementation of SkuFindSolution also
shows an example of how to use SKILL++ local functions. This usage
avoids polluting the global function space.
In the next posting of SKILL for the Skilled
we'll take a look at some shortcomings of this algorithm, particularly in regard to performance.