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# -*- coding: utf-8 -*- import random import math import re from gettext import gettext as _ import defaults
GROUP_SIZE = 9
TYPE_ROW = 0 TYPE_COLUMN = 1 TYPE_BOX = 2
digit_set = range(1, GROUP_SIZE + 1) sets = [digit_set] * 9
def is_set (row): if len(row) == len(set(row)): return True
def is_sudoku (rows): # check rows for r in rows: if not is_set(r): return False for i in range(len(rows[0])): rw = [r[i] for r in rows] if not is_set(rw): return False # check boxes width = int(math.sqrt(len(rows))) # there should be 3x3 boxes, or 4x4 if we got funky, etc. # boxes will be indices box_coordinates = [[n * width, (n + 1) * width] for n in range(width)] for x in box_coordinates: for y in box_coordinates: box = [] for xrow in [rows[ri] for ri in range(*y)]: for i in range(*x): box.append(xrow[i]) if not is_set(box): return False return True
class UnsolvablePuzzle (TypeError): pass
class ConflictError (ValueError):
def __init__ (self, conflict_type, coordinates, value): self.args = conflict_type, coordinates, value self.type = conflict_type self.coordinates = coordinates self.x = coordinates[0] self.y = coordinates[1] self.value = value
class AlreadySetError (ValueError): pass
class ParallelDict (dict): """A handy new sort of dictionary for tracking conflicts.
pd = ParallelDict() pd[1] = [2, 3, 4] # 1 is linked with 2, 3 and 4 pd -> {1:[2, 3, 4], 2:[1], 3:[1], 4:[1]} pd[2] = [1, 3, 4] # 2 is linked with 3 and 4 as well as 1 pd -> {1: [2, 3, 4], 2:[3, 4], 3:[1, 2], 4:[1, 2]} Now for the cool part... del pd[1] pd -> {2: [2, 3], 3:[2], 4:[2]}
Pretty neat, no? """ def __init__ (self, *args): dict.__init__(self, *args)
def __setitem__ (self, k, v): dict.__setitem__(self, k, set(v)) for i in v: if i == k: continue if self.has_key(i): self[i].add(k) else: dict.__setitem__(self, i, set([k]))
def __delitem__ (self, k): v = self[k] dict.__delitem__(self, k) for i in v: if i == k: continue if self.has_key(i): # Make sure we have a reference to i. If we don't # something has gone wrong... but according to bug # 385937 this has gone wrong at least once, so we'd # better check for it. if k in self[i]: self[i].remove(k) if not self[i]: # If k was the last value in the list of values # for i, then we delete i from our dictionary dict.__delitem__(self, i)
class SudokuGrid(object): def __init__ (self, grid = False, verbose = False, group_size = 9): self.grid = [] self.cols = [] self.rows = [] self.boxes = [] self.conflicts = ParallelDict() self.group_size = int(group_size) self.verbose = False self.gen_set = set(range(1, self.group_size + 1)) for n in range(self.group_size): self.cols.append(set()) self.rows.append(set()) self.boxes.append(set()) self.grid.append([0] * self.group_size) self.box_by_coords = {} self.box_coords = {} self.calculate_box_coords() # sets box_coords and box_by_coords self.row_coords = {} for n, row in enumerate([[(x, y) for x in range(self.group_size)] for y in range(self.group_size)]): self.row_coords[n] = row self.col_coords = {} for n, col in enumerate([[(x, y) for y in range(self.group_size)] for x in range(self.group_size)]): self.col_coords[n] = col if grid: if type(grid) == str: g = re.split("\s+", grid) side = int(math.sqrt(len(g))) grid = [] for row in range(side): start = row * int(side) grid.append([int(i) for i in g[start:start + side]]) self.populate_from_grid(grid) self.verbose = verbose
def calculate_box_coords (self): width = int(math.sqrt(self.group_size)) box_coordinates = [[n * width, (n + 1) * width] for n in range(width)] box_num = 0 for xx in box_coordinates: for yy in box_coordinates: self.box_coords[box_num] = [] for x in range(*xx): for y in range(*yy): self.box_by_coords[(x, y)] = box_num self.box_coords[box_num].append((x, y)) box_num += 1
def add (self, x, y, val, force = False): if not val: pass if self._get_(x, y): if force: try: self.remove(x, y) except: print 'Strange: problem with add(', x, y, val, force, ')' import traceback traceback.print_exc() else: #FIXME: This is called when the fill button #is clicked multiple times, which causes this exception: #raise AlreadySetError return # Always store the value in the underlying grid self._set_(x, y, val) # But don't add it to the solution hints(rows/cols/boxes) if there is # a conflict if val in self.rows[y]: raise ConflictError(TYPE_ROW, (x, y), val) if val in self.cols[x]: raise ConflictError(TYPE_COLUMN, (x, y), val) box = self.box_by_coords[(x, y)] if val in self.boxes[box]: raise ConflictError(TYPE_BOX, (x, y), val) # do the actual adding self.rows[y].add(val) self.cols[x].add(val) self.boxes[box].add(val)
def remove (self, x, y): val = self._get_(x, y) self.rows[y].discard(val) self.cols[x].discard(val) self.boxes[self.box_by_coords[(x, y)]].discard(val) self._set_(x, y, 0)
def _get_ (self, x, y): return self.grid[y][x]
def _set_ (self, x, y, val): self.grid[y][x] = val
def possible_values (self, x, y): return self.gen_set - self.rows[y] - self.cols[x] - self.boxes[self.box_by_coords[(x, y)]]
def pretty_print (self): print 'SUDOKU' for r in self.grid: for i in r: print i, print print
def populate_from_grid (self, grid): for y, row in enumerate(grid): for x, cell in enumerate(row): if cell: try: self.add(x, y, cell) except ConflictError: pass
def __repr__ (self): s = "<Grid\n " grid = [] for r in self.grid: grid.append(" ".join([str(i) for i in r])) s += "\n ".join(grid) return s
def calculate_open_squares (self): possibilities = {} for x in range(self.group_size): for y in range(self.group_size): if not self._get_(x, y): possibilities[(x, y)] = self.possible_values(x, y) return possibilities
def to_string (self): """Output our grid as a string.""" return " ".join([" ".join([str(x) for x in row]) for row in self.grid])
def is_valid_puzzle (p): """Check puzzle for basic validity.
This does not check for solvability or ensure a unique solution -- it merely checks well-formedness. This should provide some protection again file corruption, etc. (i.e. if we use this function to check puzzles before handing them out to the rest of the app, we'll prevent tracebacks related to corrupted puzzles). """ try: p = p.replace(' ', '') assert(len(p.replace(' ', '')) == 81) [int(c) for c in p.replace(' ', '')] except: #import traceback; traceback.print_exc() return False else: return True
def sudoku_grid_from_string (s): """Given an 81 character string, return a grid.""" s = s.replace(' ', '') assert(len(s)<=GROUP_SIZE ** 2) grid = [] i = 0 for x in range(GROUP_SIZE): row = [] for y in range(GROUP_SIZE): if len(s) <= i: n = 0 else: n = s[i] try: n = int(n) except: n = n or 0 if n in digit_set: row.append(n) else: row.append(0) i += 1 grid.append(row) return SudokuGrid(grid)
class SudokuSolver (SudokuGrid): """A SudokuGrid that can solve itself.""" def __init__ (self, grid = False, verbose = False, group_size = 9): self.current_guess = None self.initialized = False SudokuGrid.__init__(self, grid, verbose = verbose, group_size = group_size) self.virgin = SudokuGrid(grid) self.guesses = GuessList() self.breadcrumbs = BreadcrumbTrail() self.backtraces = 0 self.initialized = True self.solved = False self.solving = False self.trail = []
def auto_fill_for_xy (self, x, y): """Fill the square x,y if possible.""" possible = self.gen_set - self.rows[y] - self.cols[x] - self.boxes[self.box_by_coords[(x, y)]] if len(possible) == 1: val = possible.pop() self.add(x, y, val) return ((x, y), val) if len(possible) == 0: return -1 # check our column... for coord_set, filled_set in [(self.col_coords[x], self.cols[x]), (self.row_coords[y], self.rows[y]), (self.box_coords[self.box_by_coords[(x, y)]], self.boxes[self.box_by_coords[(x, y)]]) ]: needed_set = self.gen_set - filled_set for coord in coord_set: if self._get_(*coord): continue elif (x, y) != coord: needed_set = needed_set - self.possible_values(*coord) if needed_set and len(needed_set) == 1: val = needed_set.pop() if val in possible: self.add(x, y, val) return ((x, y), val) else: return -1 if len(needed_set)>1: return -1
def auto_fill (self): changed = [] try: changed = self.fill_must_fills() except UnsolvablePuzzle: return changed try: changed.extend(self.fill_deterministically()) finally: return changed
def fill_must_fills (self): changed = [] for label, coord_dic, filled_dic in [('Column', self.col_coords, self.cols), ('Row', self.row_coords, self.rows), ('Box', self.box_coords, self.boxes)]: for n, coord_set in coord_dic.items(): skip_set = False for coord in coord_set: if self.conflicts.has_key(coord): skip_set = True break if skip_set: continue needs = dict([(n, False) for n in range(1, self.group_size + 1)]) for coord in coord_set: val = self._get_(*coord) if val: # We already have this value set... if needs.has_key(val): del needs[val] else: # Otherwise, register ourselves as possible # for each number we could be for v in self.possible_values(*coord): # if we don't yet have a possible number, plug ourselves in if needs.has_key(v): if not needs[v]: needs[v] = coord else: del needs[v] for n, coords in needs.items(): if not coords: raise UnsolvablePuzzle('Missing a %s in %s' % (n, label)) else: try: self.add(coords[0], coords[1], n) changed.append((coords, n)) except AlreadySetError: raise UnsolvablePuzzle( "%s,%s must be two values at once!" % (coords) ) return changed
def fill_deterministically (self): poss = self.calculate_open_squares().items() one_choice = filter(lambda x: len(x[1]) == 1, poss) retval = [] for coords, choices in one_choice: if self.verbose: print 'Deterministically adding ', coords, choices val = choices.pop() self.add(coords[0], coords[1], val) retval.append([(coords[0], coords[1]), val]) if self.verbose: print 'deterministically returning ', retval return retval
def solve (self): if self.solving: return self.solving = True self.auto_fill() while not self.guess_least_open_square(): pass if self.verbose: print 'Solved!\n', self self.solving = False self.solved = True
def solution_finder (self): self.auto_fill() while not self.guess_least_open_square(): pass self.solved = True yield tuple([tuple(r) for r in self.grid[0:]]) while self.breadcrumbs: self.unwrap_guess(self.breadcrumbs[-1]) try: while not self.guess_least_open_square(): pass except UnsolvablePuzzle: break else: yield tuple([tuple(r) for r in self.grid[0:]]) yield None
def has_unique_solution (self): sf = self.solution_finder() sf.next() if sf.next(): return False else: return True
def guess_least_open_square (self): # get open squares and check them poss = self.calculate_open_squares().items() # if there are no open squares, we're done! if not poss: if self.verbose: print 'Solved!' return True # otherwise, find the possibility with the least possibilities poss.sort(lambda a, b: len(a[1]) > len(b[1]) and 1 or len(a[1]) < len(b[1]) and -1 or \ a[0] > b[0] and 1 or a[1] < b[1] and -1 or 0) least = poss[0] # remove anything we've already guessed possible_values = least[1] - self.guesses.guesses_for_coord(*least[0]) if not possible_values: if self.breadcrumbs: self.backtraces += 1 self.unwrap_guess(self.breadcrumbs[-1]) return self.guess_least_open_square() else: raise UnsolvablePuzzle("Unsolvable %s.\n \ Out of guesses for %s. Already guessed\n \ %s (other guesses are %s)" % (self, least[0], self.guesses.guesses_for_coord(*least[0]), self.guesses)) guess = random.choice(list(possible_values)) # Create guess object guess_obj = Guess(least[0][0], least[0][1], guess) if self.breadcrumbs: self.breadcrumbs[-1].children.append(guess_obj) self.current_guess = None #reset (we're tracked via guess.child) self.add(least[0][0], least[0][1], guess) self.current_guess = guess_obj # (All deterministic additions # get added to our # consequences) self.guesses.append(guess_obj) self.trail.append(('+', guess_obj)) self.breadcrumbs.append(guess_obj) try: self.auto_fill() except NotImplementedError: self.trail.append('Problem filling coordinates after guess') self.unwrap_guess(guess_obj) return self.guess_least_open_square() if set([]) in self.calculate_open_squares().values(): self.trail.append('Guess leaves us with impossible squares.') self.unwrap_guess(guess_obj) return self.guess_least_open_square()
def unwrap_guess (self, guess): self.trail.append(('-', guess)) if self._get_(guess.x, guess.y): self.remove(guess.x, guess.y) for consequence in guess.consequences.keys(): if self._get_(*consequence): self.remove(*consequence) for child in guess.children: self.unwrap_guess(child) if child in self.guesses: self.guesses.remove(child) if guess in self.breadcrumbs: self.breadcrumbs.remove(guess)
def pad (self, n, pad_to): n = str(n) padding = int(pad_to) - len(n) second_half = padding / 2 first_half = second_half + padding % 2 return " " * first_half + n + " " * second_half
def add (self, x, y, val, *args, **kwargs): if self.current_guess: self.current_guess.add_consequence(x, y, val) SudokuGrid.add(self, x, y, val, *args, **kwargs)
class InteractiveSudoku (SudokuSolver): """A subclass of SudokuSolver that provides some convenience functions for helping along a human.who is in the midst of solving.""" def __init__ (self, grid = False, verbose = False, group_size = 9): SudokuSolver.__init__(self, grid, verbose, group_size) self.cleared_conflicts = []
def to_string (self): return self.virgin.to_string() + '\n' + SudokuSolver.to_string(self)
def find_impossible_implications (self, x, y): """Return a list of impossibilities implied by the users actions.""" row_cells = self.row_coords[y] col_cells = self.col_coords[x] box = self.box_by_coords[(x, y)] box_cells = self.box_coords[box] for coord_set in [row_cells, col_cells, box_cells]: broken = [] # just work on the open squares coord_set = filter(lambda coords: not self._get_(*coords), coord_set) for coords in coord_set: if not self.possible_values(*coords): broken.append(coords) return broken
def check_for_completeness (self): for r in self.rows: if len(r) != self.group_size: return False for c in self.cols: if len(c) != self.group_size: return False return True
def is_changed (self): return (self.grid != self.virgin.grid)
def add (self, x, y, val, force = False): '''Add a value to the grid.
The main feature of this method is conflict resolution. When conflicts are found they are stored in the conflicts ParallelDict. A cell that is in conflict is stored in the underlying grid(SudokuGrid.grid), but it has all of its solution hints cleared(SudokuGrid.rows/cols/boxes). Care must be taken so that solution hints from the original grid(SudokuSolver.virgin) are not cleared. ''' # First just add it to SudokuGrid no_exception = True try: super(InteractiveSudoku, self).add(x, y, val, force) except ConflictError: no_exception = False
# Find any cells that conflict with the new value for this cell coords = set([]) coords.update(self.row_coords[y]) coords.update(self.col_coords[x]) coords.update(self.box_coords[self.box_by_coords[(x, y)]]) coords.discard((x, y)) conflicting_coordinates = [] for xx, yy in coords: if self._get_(xx, yy) == val: conflicting_coordinates.append((xx, yy)) # Store the conflicts for access if conflicting_coordinates: self.conflicts[(x, y)] = conflicting_coordinates # Resume when there are no conflicts else: return # But when we do have conflicts, the values from cols/rows/boxes need # to be removed so the hinting doesn't consider them. We must be # chaste with the virgin though. try: if no_exception and not self.virgin._get_(x, y): self.rows[y].discard(val) self.cols[x].discard(val) self.boxes[self.box_by_coords[(x, y)]].discard(val) for xx, yy in conflicting_coordinates: if self.virgin._get_(xx, yy): continue if not val in self.virgin.rows[yy]: self.rows[yy].discard(val) if not val in self.virgin.cols[xx]: self.cols[xx].discard(val) if not val in self.virgin.box_coords[self.box_by_coords[(xx, yy)]]: self.boxes[self.box_by_coords[(xx, yy)]].discard(val) # This class can be used before the virgin is created. Pass through # for the initialization phase except AttributeError: pass
def remove (self, x, y): '''Remove a value from the grid.
The main feature of this method is conflict resolution. All conflicting cells are checked to see if they are actually conflict-free. A list of conflict-free cells are stored in InteractiveSudoku.cleared_conflicts. The cleared_conflicts list is cleared for each meaningful call to remove(), so it must be processed before another remove() call. All solution hints(SudokuGrid.rows/cols/boxes) are reinstated for conflict-free cells. ''' # Grab the value that we're clearing. Skip out if its nothing val = self._get_(x, y) if not val: return # Pop the conflicts resolved by this removal self.cleared_conflicts = [] errors_removed = [] if self.conflicts.has_key((x, y)): errors_removed = self.conflicts[(x, y)] del self.conflicts[(x, y)] # If there are no conflicts for this cell then just remove it in from # the grid else : super(InteractiveSudoku, self).remove(x, y) return # Grid clearance flags if val in self.rows[y]: clear_row = True else: clear_row = False if val in self.cols[x]: clear_col = True else: clear_col = False if val in self.boxes[self.box_by_coords[(x, y)]]: clear_box = True else: clear_box = False # Scroll through the conflicts for coord in errors_removed: # If it is not an error by some other pairing, append it to a list # of conflicts that were actually cleared by this removal. if not self.conflicts.has_key(coord): self.cleared_conflicts.append(coord) # When a conflict remains, we need to correct the rows, cols, and # boxes arrays properly else: if clear_row and coord in self.row_coords[y]: clear_row = False if clear_col and coord in self.col_coords[x]: clear_col = False if clear_box and coord in self.box_coords[self.box_by_coords[(x, y)]]: clear_box = False
# Clear the rows, cols, and boxes if we need to. if clear_row: self.rows[y].remove(val) if clear_col: self.cols[x].remove(val) if clear_box: self.boxes[self.box_by_coords[(x, y)]].remove(val) # Clear the cell self._set_(x, y, 0)
# Scroll through the cleared conflicts and commit them to ensure they # are represented in the grid properly. It is possible for add() to do # subsequent remove()s, so hold onto the cleared conflict list for the # caller. hold_conflicts = self.cleared_conflicts for xx, yy in self.cleared_conflicts: self.add(xx, yy, val, True) self.cleared_conflicts = hold_conflicts
class DifficultyRating:
very_hard = _('Very Hard') hard = _('Hard') medium = _('Medium') easy = _('Easy')
very_hard_range = (0.75, 10) hard_range = (0.6, 0.75) medium_range = (0.45, 0.6) easy_range = (-10, 0.45)
categories = {'very hard':very_hard_range, 'hard':hard_range, 'medium':medium_range, 'easy':easy_range}
ordered_categories = ['easy', 'medium', 'hard', 'very hard'] label_by_cat = {'easy':easy, 'medium':medium, 'hard':hard, 'very hard':very_hard}
def __init__ (self, fill_must_fillables, elimination_fillables, guesses, backtraces, squares_filled): self.fill_must_fillables = fill_must_fillables self.elimination_fillables = elimination_fillables self.guesses = guesses self.backtraces = backtraces self.squares_filled = squares_filled if self.fill_must_fillables: self.instant_fill_fillable = float(len(self.fill_must_fillables[0])) else: self.instant_fill_fillable = 0.0 if self.elimination_fillables: self.instant_elimination_fillable = float(len(self.elimination_fillables[0])) else: self.instant_elimination_fillable = 0.0
self.proportion_instant_elimination_fillable = self.instant_elimination_fillable / self.squares_filled # some more numbers that may be crazy... self.proportion_instant_fill_fillable = self.instant_fill_fillable / self.squares_filled self.elimination_ease = add_with_diminishing_importance( self.count_values(self.elimination_fillables) ) self.fillable_ease = add_with_diminishing_importance( self.count_values(self.fill_must_fillables) ) self.value = self.calculate()
def count_values (self, dct): kk = dct.keys() kk.sort() return [len(dct[k]) for k in kk]
def calculate (self): return 1 - float(self.fillable_ease) / self.squares_filled \ - float(self.elimination_ease) / self.squares_filled \ + len(self.guesses) / self.squares_filled \ + self.backtraces / self.squares_filled
def __repr__ (self): return '<DifficultyRating %s>' % self.value
def pretty_print (self): for name, stat in [('Number of moves instantly fillable by elimination', self.instant_elimination_fillable), ('Percentage of moves instantly fillable by elimination', self.proportion_instant_elimination_fillable * 100), ('Number of moves instantly fillable by filling', self.instant_fill_fillable), ('Percentage of moves instantly fillable by filling', self.proportion_instant_fill_fillable * 100), ('Number of guesses made', len(self.guesses)), ('Number of backtraces', self.backtraces), ('Ease by filling', self.fillable_ease), ('Ease by elimination', self.elimination_ease), ('Calculated difficulty', self.value) ]: print name, ': ', stat
def value_string (self): if self.value > self.very_hard_range[0]: return _(self.very_hard) elif self.value > self.hard_range[0]: return _(self.hard) elif self.value > self.medium_range[0]: return _(self.medium) else: return _(self.easy)
def value_category (self): """Get category string, without i18n or capitalization
For use in categorizing category. """ if self.value > self.very_hard_range[0]: return 'very hard' elif self.value > self.hard_range[0]: return 'hard' elif self.value > self.medium_range[0]: return 'medium' else: return 'easy'
def get_difficulty_category_name (diff_float): return DifficultyRating.label_by_cat.get( get_difficulty_category(diff_float), '?' )
def get_difficulty_category (diff_float): for category, range in DifficultyRating.categories.items(): if range[0] <= diff_float < range[1]: return category
class SudokuRater (SudokuSolver):
def __init__ (self, grid = False, verbose = False, group_size = 9): self.initialized = False self.guessing = False self.fake_add = False self.fake_additions = [] self.filled = set([]) self.fill_must_fillables = {} self.elimination_fillables = {} self.tier = 0 SudokuSolver.__init__(self, grid, verbose, group_size)
def add (self, *args, **kwargs): if not self.fake_add: if self.initialized and not self.guessing: self.scan_fillables() for delayed_args in self.add_me_queue: coords = (delayed_args[0], delayed_args[1]) if not self._get_(*coords): SudokuSolver.add(self, *delayed_args) if not self._get_(args[0], args[1]): SudokuSolver.add(self, *args) self.tier += 1 else: SudokuSolver.add(self, *args, **kwargs) else: self.fake_additions.append(args)
def scan_fillables (self): self.fake_add = True # this will now tell us how many squares at current # difficulty could be filled at this moment. self.fake_additions = [] try: self.fill_must_fills() except: pass self.fill_must_fillables[self.tier] = set(self.fake_additions[:]) - self.filled self.add_me_queue = self.fake_additions[:] self.fake_additions = [] try: self.fill_deterministically() except: pass self.elimination_fillables[self.tier] = set(self.fake_additions[:]) - self.filled self.filled = self.filled | self.fill_must_fillables[self.tier] | self.elimination_fillables[self.tier] self.add_me_queue.extend(self.fake_additions[:]) self.fake_add = False
def guess_least_open_square (self): self.guessing = True return SudokuSolver.guess_least_open_square(self)
def difficulty (self): if not self.solved: self.solve() self.clues = 0 # Add up the number of our initial clues through some nifty mapping calls map(lambda r: map(lambda i: setattr(self, 'clues', self.clues.__add__(i and 1 or 0)), r), self.virgin.grid) self.numbers_added = self.group_size ** 2 - self.clues rating = DifficultyRating(self.fill_must_fillables, self.elimination_fillables, self.guesses, self.backtraces, self.numbers_added) return rating
class GuessList (list): def __init__ (self, *guesses): list.__init__(self, *guesses)
def guesses_for_coord (self, x, y): return set([guess.val for guess in filter(lambda guess: guess.x == x and guess.y == y, self)])
def remove_children (self, guess): removed = [] for g in guess.children: if g in self: removed.append(g) self.remove(g) return removed
def remove_guesses_for_coord (self, x, y): nuking = False nuked = [] for i in range(len(self) - 1): g = self[i - len(nuked)] if g.x == x and g.y == y: nuking = True if nuking: self.remove(g) nuked += [g] return nuked
class BreadcrumbTrail (GuessList): def append (self, guess): # Raise an error if we add something to ourselves twice if self.guesses_for_coord(guess.x, guess.y): raise ValueError("We already have crumbs on %s, %s" % (guess.x, guess.y)) else: list.append(self, guess)
class Guess: def __init__ (self, x, y, val): self.x = x self.y = y self.children = [] self.val = val self.consequences = {}
def add_consequence (self, x, y, val): self.consequences[(x, y)] = val
def __repr__ (self): s = "<Guess (%s, %s)=%s" % (self.x, self.y, self.val) if self.consequences: s += " implies: " s += ", ".join(["%s->%s" % (k, v) for k, v in self.consequences.items()]) s += ">" return s
def add_with_diminishing_importance (lst, diminish_by = lambda x: x + 1): sum = 0 for i, n in enumerate(lst): sum += float(n) / diminish_by(i) return sum
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