I've written a simple KenKen puzzle/solver in Python. I'd love some feedback on the design of the puzzle, as well as the architecture for the solver.To model the puzzle, I have the following classes:Cell is used to model a (row, col, value) tupleCage (abstract) is used to model a grouping of Cell objects that must collectively satisfy a constraint. From this class, we have the following derived classes:AddCage for cells involved in addition constraintsMulCage for cells involved in multiplication constraintsSubCage for cells involved in subtraction constraintsDivCage for cells involved in division constraintsConCage for constant constraintsRowCage for unique row/column constraintsPuzzle combines cages, cells, and exposes methods for the unassigned cells, whether or not the puzzle is solved, etc.Now for the code:from abc import ABC, abstractmethodfrom utils import kk_add, kk_mul, kk_sub, kk_divclass Cell:    def __init__(self, row, col, value=None):                Models a cell in a kenken puzzle        Args:            row: row            col: column            value: cell value                self.row = row        self.col = col        self.value = value    def __str__(self):        return '<Cell ({0}, {1}): {2}>'.format(self.row, self.col, self.value)    def __hash__(self):        return hash((self.row, self.col))    def accept(self, visitor):                Visitor implementation; accept a visitor object        and call the object's visit method for this object        Args:            visitor: `CellVisitor` implementation         Returns: None                visitor.visit_cell(self)class Cage(ABC):    def __init__(self, cells, func):                Base class to model a cage in a kenken puzzle        A cage is a grouping of cells with a constraint        that the values of the cells must collectively satisfy        Args:            cells: the `Cell` objects in this cage            func: a predicate used to indicate when the cage is satisfied                self.cells = set(cells)        self.func = func    def __str__(self):        return '<{0} cells={1}>'.format(self.__class__.__name__, self.cells)    @property    def values(self):                 Returns: list the cell values list for this cage                return [cell.value for cell in self.cells]    @property    def consistent(self):                Returns: bool whether or not this cage is consistent        with respect to its current cell values                return None in self.values or self.solved    @property    def solved(self):                Returns: bool whether or not this cage is solved        with respect to its current cell values                values = self.values        return (            None not in values            and len(values) == len(self.cells)            and self.evaluate(*values)        )    def evaluate(self, *values):                Evaluate this cage for the given input arguments,        returning whether or not it's conditions have been satisfied        Args:            *values: variate list of test values        Returns: bool                return self.func(values)    @abstractmethod    def accept(self, visitor):                Visit this cage. Accept a visitor object and call the        object's visit method for this object        Args:            visitor: `CageVisitor` implementation         Returns: None                passclass AddCage(Cage):    def __init__(self, cells, value):                Models an addition cage in a kenken puzzle        Args:            cells: list of `Cell` objects contained in this cage            value: target value the cell values in this cage must sum to                self.value = value        super().__init__(cells, lambda values: kk_add(values, value))    def accept(self, visitor):                Visit this cage        Args:            visitor: `CageVisitor` object        Returns: None                visitor.visit_add(self)class MulCage(Cage):    def __init__(self, cells, value):                Models a multiplication cage in a kenken puzzle        Args:            cells: list of `Cell` objects contained in this cage            value: target value the cell values in this cage must multiply to                self.value = value        super().__init__(cells, lambda values: kk_mul(values, value))    def accept(self, visitor):                Visit this cage        Args:            visitor: `CageVisitor` object        Returns: None                visitor.visit_mul(self)class SubCage(Cage):    def __init__(self, cells, value):                Models a subtraction cage in a kenken puzzle        Args:            cells: list of `Cell` objects contained in this cage            value: target value the cell values in this cage must subtract to                self.value = value        super().__init__(cells, lambda values: kk_sub(values, value))    def accept(self, visitor):                Visit this cage        Args:            visitor: `CageVisitor` object        Returns: None                visitor.visit_sub(self)class DivCage(Cage):    def __init__(self, cells, value):                Models a division cage in a kenken puzzle        Args:            cells: list of `Cell` objects contained in this cage            value: target value the cell values in this cage must divide to                self.value = value        super().__init__(cells, lambda values: kk_div(values, value))    def accept(self, visitor):                Visit this cage        Args:            visitor: `CageVisitor` object        Returns: None                visitor.visit_div(self)class ConCage(Cage):    def __init__(self, cell, value):                Models a constant cage in a kenken puzzle        Args:            cell: `Cell` object for this cage            value: target value                def func(values):            return len(values) == 1 and values[0] == value        self.value = value        super().__init__([cell], func)    def accept(self, visitor):                Visit this cage        Args:            visitor: `CageVisitor` object        Returns: None                visitor.visit_con(self)class RowCage(Cage): # RowConstraint    def __init__(self, cells):                Models a row constraint in a kenken puzzle        Here the cell values in this cage must be all unique        for the cage to be solved        Args:            cells: `Cell` objects                def func(values):            return len(values) == len(set(values))        super().__init__(cells, func)    def accept(self, visitor):                Visit this cage        Args:            visitor: `CageVisitor` object        Returns: None                visitor.visit_row(self)class Puzzle:    def __init__(self, width, cells, cages):                Models a kenken puzzle        See https://en.wikipedia.org/wiki/KenKen        for more information        Args:            width: puzzle size            cells: `Cell` objects comprising this puzzle            cages: `Cage` objects a solution for this puzzle must satisfy                self.width = width        self.cells = cells        self.cages = cages    def __str__(self):        return '<Puzzle width={0}, cages={1}>'.format(            self.width, len(self.cages)        )    @property    def domain(self):                Returns: bool this puzzle's possible cells values                return range(1, self.width + 1)    @property    def unassigned(self):                Returns: bool this puzzle's unassigned cells                return (cell for cell in self.cells if cell.value is None)    @property    def solved(self):                Returns: bool whether or not this puzzle has been solved                return all(cage.solved for cage in self.cages)    def consistent(self, cell):                Returns whether or not the value for the given cell is consistent        with all of its cage constraints        Args:            cell: `Cell` object        Returns: bool                return all(cage.consistent for cage in self.cages if cell in cage.cells)For both the Cell and the Cage classes, we have an accept method. This is used to make the objects amenable to the visitor design pattern, for use during solving. The idea is that each cell has a set of candidate values that needs to be reduced after we decide to place a value for the cell. I decided to expose things this way to make edits to the core puzzle logic less frequent. Moreover, to try different solution strategies, we need only change the implementation of the visitor we pass to the cells/cages; the core puzzle components need not be changed.Let's look at the solver classes:CellVisitor is used to visit cellsCageVisitor is used to visit cages; its lifetime is managed by a CellVisitorAnd the code:from utils import with_timing, kk_div, kk_subclass CellVisitor:    def __init__(self, candidates, cages):                Visitor for puzzle cells        Pass an instance of this object to a puzzle cell        to visit the cell and all the cages that involve        this cell        Here we use this object to model the process of eliminating        a set of candidate values for the given cell        See https://en.wikipedia.org/wiki/Visitor_pattern        for more information on this design pattern        Args:            candidates: list of cell candidates            cages: list of cages this visitor should also visit                self.candidates = candidates        self.cages = cages    def __str__(self):        return '<CellVisitor candidates={0}>'.format(self.candidates)    def visit_cell(self, cell):                Visits a `Cell`        Visit each cage that contains this cell; the resulting        candidates will be the possible values for this cell        Args:            cell: `Cell` object to visit        Returns: None                visitor = CageVisitor(self.candidates)        for cage in self.cages:            cage.accept(visitor)class CageVisitor:    def __init__(self, candidates):                Visitor for puzzle cages        The methods in this object are used to prune the cell        candidate values        Args:            candidates: cell candidate values to prune                self.candidates = candidates    def __str__(self):        return '<CageVisitor candidates={0}>'.format(self.candidates)    def visit_add(self, cage):                Visits an `AddCage`        We start with the current cage sum. Any        value that exceeds the cage target value is pruned        Args:            cage: `AddCage` object to visit        Returns: None                s = sum(value for value in cage.values if value)        for value in self.candidates[:]:            if value + s > cage.value:                self.prune(value)    def visit_mul(self, cage):                Visits a `MulCage`        Any candidate value that is not a divisor of        the cage target value is pruned        Args:            cage: `MulCage` object to visit        Returns: None                for value in self.candidates[:]:            if cage.value % value != 0:                self.prune(value)    def visit_sub(self, cage):                Visits a `SubCage`        This implementation removes pairs from the        candidates if the difference of a given pair        is not equal to the cage value        Args:            cage: `MulCage` object to visit        Returns: None                candidates = self.candidates[:]        for x in candidates:            if not any(kk_sub([x, y], cage.value) for y in candidates):                self.prune(x)    def visit_div(self, cage):                Visits a `DivCage`        This implementation removes pairs from the        candidates if the quotient of a given pair        is not equal to the cage value        Args:            cage: `DivCage` object to visit        Returns: None                candidates = self.candidates[:]        for x in candidates:            if not any(kk_div([x, y], cage.value) for y in candidates):                self.prune(x)    def visit_con(self, cage):                Visits a `ConCage`        This implementation removes all candidates        that are not equal to the cage target value        Args:            cage: `ConCage` object to visit        Returns: None                for x in self.candidates[:]:            if x != cage.value:                self.prune(x)    def visit_row(self, cage):                Visits a `RowCage`        This implementation removes all values        that are already assigned to a cell in the row        Args:            cage: `ConCage` object to visit        Returns: None                for value in cage.values:            self.prune(value)    def prune(self, value):                Helper method to safely remove values        from the candidates        Args:            value: to remove        Returns: None                if value in self.candidates:            self.candidates.remove(value)@with_timingdef backtrack_solve(puzzle):        Solves a kenken puzzle recursively    During each iteration of the algorithm, a filtering    strategy is applied to the puzzle's remaining unassigned cells    See https://en.wikipedia.org/wiki/Backtracking    for more information on this algorithm    Args:        puzzle: `Puzzle` object to solve    Returns: bool True if all values in `puzzle` have been assigned a value        def reduce(cell):                Reduce the candidate values for this cell        Args:            cell: `Cell` object to reduce        Returns: list of reduced candidates                candidates = list(puzzle.domain)        cages = (cage for cage in puzzle.cages if cell in cage.cells)        cell.accept(CellVisitor(candidates, cages))        return candidates    def solve():                Solve this puzzle recursively        The algorithm first reduces the candidates for the puzzle's        unassigned cells        We then sort the reduced cells by candidate length and        recursively try values for the current cell until the search        successfully solves the puzzle        Returns: bool                reduced = {cell: reduce(cell) for cell in puzzle.unassigned}        for cell in sorted(reduced, key=lambda c: len(reduced[c])):            for value in reduced[cell]:                cell.value = value                if puzzle.consistent(cell):                    if solve():                        return True                cell.value = None            return False        return puzzle.solved    return solve()You can read more about the algorithm in the documentation for the solver. The basic idea is that when we visit a cell, we start off with the puzzle's full domain. Each of the cages reduces the candidates further, by means of a filtering strategy that is invoked on the candidates when we visit that cage. We do this reduce operation for each of the unassigned cells.Finally, I have a utils.py that contains definitions that are in use by the solver and puzzle files. Included is a parse_string method that can be used to create a Puzzle object from a dictionary string:import timefrom ast import literal_evalfrom functools import wraps, reducedef kk_add(values, value):        Returns whether or not the given values    sum to the target value    Args:        values: list of test values        value: target value    Returns: bool        return sum(values) == valuedef kk_mul(values, value):        Returns whether or not the given values    multiply to the target value    Args:        values: list of test values        value: target value    Returns: bool        return product(values) == valuedef kk_sub(values, value):        Returns whether or not the given values subtract    to the target value    Args:        values: list of test values        value: target value    Returns: bool        return abs(values[0] - values[1]) == valuedef kk_div(values, value):        Returns whether or not the given values divide    to the target value    Args:        values: list of test values        value: target value    Returns: bool        return (int(values[0] / values[1]) == value or            int(values[1] / values[0]) == value)def product(nums):        Helper method to compute the product of a list    of numbers    Args:        nums: list of numbers    Returns: number        return reduce(lambda x, y: x * y, nums, 1)def with_timing(f, output=print):        Helper method to run a function and output    the function run time    Args:        f: function to decorate        output: function to output the time message    Returns: callable decorated function        @wraps(f)    def timed(*args, **kwargs):        ts = time.time()        result = f(*args, **kwargs)        te = time.time()        message = 'func:{!r} args:[{!r}, {!r}] took: {:2.4f} sec'.format(            f.__name__, args, kwargs, te - ts        )        output(message)        return result    return timeddef parse_string(s):        Parse a string to a `Puzzle` object    The string should be a dictionary that python    can interpret literally. For example:    {      'size': 2,       'cages': [         {'value': 2, 'op': '/', 'cells': [(0,0), (0,1)]},         {'value': 3, 'op': '+', 'cells': [(1,0), (1,1)]}      ]    }    The 'op' should be one of :        '+' -> AddCage,        '-' -> SubCage,        '*' -> MulCage,        '/' -> DivCage,        '$' -> ConCage    The exclusive row/column cages will be created automatically    Args:        s: input string to read    Returns: `Puzzle` object        from puzzle import (        Cell,        AddCage,        SubCage,        MulCage,        DivCage,        ConCage,        RowCage,        Puzzle    )    d = literal_eval(s.strip())    cage_factory = {        '+': AddCage,        '-': SubCage,        '*': MulCage,        '/': DivCage,        '$': ConCage    }    size = d.get('size')    cages = d.get('cages')    if size is None or cages is None:        raise SyntaxError(            Expected 'size' and 'cages'. Got `{0}`.format(d)        )    puzzle_cages = []    puzzle_cells = set()    for cage in cages:        value = cage.get('value')        cells = cage.get('cells')        if any(cell in puzzle_cells for cell in cells):            raise ValueError('Some cells exist in another cage {0}'.format(cells))        if not value or not cells:            raise SyntaxError(                Expected 'value' and 'cells'. Got {0}.format(cage)            )        op = cage.get('op')        if op not in cage_factory:            raise SyntaxError(                Expected '+', '-', '*', '/', '$'. Got {0}.format(op)            )        if op == '$' and len(cells) > 1:            raise ValueError(Expected one cell for `ConstantConstraint`)        cage_cells = []        for (row, col) in cells:            cell = Cell(row, col, None)            cage_cells.append(cell)        puzzle_cells = puzzle_cells.union(cage_cells)        # the constructor for `ConCage` expects a single cell as oppose to a list        cage = cage_factory[op](cage_cells[0] if op == '$' else cage_cells, value)        puzzle_cages.append(cage)    if len(puzzle_cells) != size * size:        raise Exception(            'Expected {0} cells; parsed {1}'.format(                size*size, puzzle_cells)        )    for row in range(size):        cells = [cell for cell in puzzle_cells if cell.row == row]        puzzle_cages.append(RowCage(cells))    for col in range(size):        cells = [cell for cell in puzzle_cells if cell.col == col]        puzzle_cages.append(RowCage(cells))    return Puzzle(size, puzzle_cells, puzzle_cages)Any feedback is welcome. I have some additional puzzle files that I used while debugging/testing the solving algorithm, as well as a run.py file which provides a CLI for this application. If you think this is needed, feel free to leave a comment and I can provide a link.
A KenKen puzzle/solver in Python
python;design patterns;recursion;backtracking;visitor pattern
I am unfamiliar with both the design pattern and the backtracking algorithm, so almost all of my comments will be nitpicks.utils.pykk_sub: I prefer values[0] - values[1] in {value, -value}, which will work if value is negative. Maybe we know that it will always be nonnegative, though.kk_div: I prefer values[0] == values[1] * value or values[1] == values[0] * value, which avoids float division and casts.kk_{add,mul,sub,div}: It would be clearer to me if you used more descriptive argument names than 'values' and 'value'. You could do 'summands' and 'sum' (though that clashes with the built-in) for kk_add, 'factors' and 'product' for kk_mul, etc.product: I prefer reduce(operator.mul, values). You're already using functools, so I don't feel that this is a big jump in complexity.parse_string:It would be clearer to me if you used a set comprehension to define cage_cells: {Cell(row, col, None) for (row, col) in cells}. Also, since value is a keyword argument in Cell.__init__ you could use set(itertools.starmap(Cell, cells)) (not that that's any clearer than what you have now).If you moved that definition up to right after the definition of cells, you could replace the test for already-accounted-for cells with if puzzle_cells.intersection(cage_cells), which is easier for me to parse.I think you should swap the test for already-accounted-for cells with the test for value or cells not being provided. It seems strange to use cells and only afterwards check that it's valid.You could avoid the searches in the collection of the cells in each row and column using something like below.rows = [[] for _ in range(size)]cols = [[] for _ in range(size)]# Later, inside look over cages:for cell in cage_cells:    rows[cell.row].append(cell)    cols[cell.col].append(cell)# Then, outside that loop:puzzle_cages.extend(map(RowCage, itertools.chain(rows, cols)))class definitionsCage.solved: I think the test that values and self.cells have the same length should either be removed or throw an exception if it fails.  values is freshly created from self.cells immediately before the test; something catastrophic has happened if it fails.{Add,Mul,Sub,Div}Cage.__init__: if you switched the order of the arguments in kk_{add,mul,sub,div}, the super().__init__ call could use functools.partial(kk_{add,mul,sub,div}, value). I personally prefer that to the lambda, though I'd believe others would disagree with me. Also (this is out of my depth), doesn't this approach result in every instance of Cage having a dedicated func function object? It seems that all the instances of AddCage could share kk_add and just pass it both value and values (and likewise for the others). I think the subclasses of Cage should override evaluate, or func should be made a method.Puzzle.__init__: the puzzle comprises the cells, not the other way around.Puzzle.{domain,unassigned}: the docstring incorrectly says that a Boolean is returned.Puzzle.consistent: you repeat this expression for cages containing a given cell in reduce. Maybe it would be good to make a method for this?solver codeCageVisitor.visit_add:I'd prefer if value is not None in place of if value. Seeing if value along with sum makes me think of zero.If you found the difference between cell.value and s before the loop, you could avoid the additions inside the loop.You could prune more aggressively by factoring in how many other cells in the cage are undetermined.  For example, if cage.value is 9, cage.values is [3, None, None], and self.candidates is [1, 4, 6], we can eliminate 6 because that would force the remaining undetermined cell to have value 0. That does couple the candidates for different cells, which it looks like you haven't done so far.CageVisitor.visit_mul: I prefer if cage.value % value. And why not use the quotient of cage.value and the product of the values of the cells whose values have been determined?CageVisitor.visit_sub:In the docstring, 'MulCage' should be 'SubCage'.I don't like how kk_sub([x, x], cage.value) will get called, and you aren't using the fact that the order of values doesn't affect kk_sub. Maybe you could do something like below. You could do something similar in visit_div.self.candidates = list(itertools.chain.from_iterable(filter(    lambda pair: kk_sub(pair, cage.value),    itertools.combinations(candidates, 2))))But that said, why are you finding differences between elements of candidates? It's the difference between x and the candidates of the other cells that matters, unless I misunderstand something. It seems to me that CageVisitor is initialized with a list of candidate values for the particular cell being visited, not a list of candidate values across all cells in the cage. I may well be confused on this point.Shouldn't we make use of the cells whose values have already been determined, as is done in CageVisitor.visit_add?CageVisitor.visit_con: I may be missing something, but why not just do self.candidates = [cage.value]?CageVisitor.prune: why not make self.candidates a set or something else with faster removals than a list? I may be missing something tricky about iterating over the structure while modifying it.