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The Algorithms

Sudoku Solver

Please do not modify this file!  It is published at with
only minimal changes to work with modern versions of Python.  If you have improvements,
please make them in a separate file.

import random
import time

def cross(items_a, items_b):
    "Cross product of elements in A and elements in B."
    return [a + b for a in items_a for b in items_b]

digits = "123456789"
rows = "ABCDEFGHI"
cols = digits
squares = cross(rows, cols)
unitlist = (
    [cross(rows, c) for c in cols]
    + [cross(r, cols) for r in rows]
    + [cross(rs, cs) for rs in ("ABC", "DEF", "GHI") for cs in ("123", "456", "789")]
units = {s: [u for u in unitlist if s in u] for s in squares}
peers = {s: set(sum(units[s], [])) - {s} for s in squares}  # noqa: RUF017

def test():
    "A set of unit tests."
    assert len(squares) == 81
    assert len(unitlist) == 27
    assert all(len(units[s]) == 3 for s in squares)
    assert all(len(peers[s]) == 20 for s in squares)
    assert units["C2"] == [
        ["A2", "B2", "C2", "D2", "E2", "F2", "G2", "H2", "I2"],
        ["C1", "C2", "C3", "C4", "C5", "C6", "C7", "C8", "C9"],
        ["A1", "A2", "A3", "B1", "B2", "B3", "C1", "C2", "C3"],
    # fmt: off
    assert peers["C2"] == {
        "A2", "B2", "D2", "E2", "F2", "G2", "H2", "I2", "C1", "C3",
        "C4", "C5", "C6", "C7", "C8", "C9", "A1", "A3", "B1", "B3"
    # fmt: on
    print("All tests pass.")

def parse_grid(grid):
    """Convert grid to a dict of possible values, {square: digits}, or
    return False if a contradiction is detected."""
    ## To start, every square can be any digit; then assign values from the grid.
    values = {s: digits for s in squares}
    for s, d in grid_values(grid).items():
        if d in digits and not assign(values, s, d):
            return False  ## (Fail if we can't assign d to square s.)
    return values

def grid_values(grid):
    "Convert grid into a dict of {square: char} with '0' or '.' for empties."
    chars = [c for c in grid if c in digits or c in "0."]
    assert len(chars) == 81
    return dict(zip(squares, chars))

def assign(values, s, d):
    """Eliminate all the other values (except d) from values[s] and propagate.
    Return values, except return False if a contradiction is detected."""
    other_values = values[s].replace(d, "")
    if all(eliminate(values, s, d2) for d2 in other_values):
        return values
        return False

def eliminate(values, s, d):
    """Eliminate d from values[s]; propagate when values or places <= 2.
    Return values, except return False if a contradiction is detected."""
    if d not in values[s]:
        return values  ## Already eliminated
    values[s] = values[s].replace(d, "")
    ## (1) If a square s is reduced to one value d2, then eliminate d2 from the peers.
    if len(values[s]) == 0:
        return False  ## Contradiction: removed last value
    elif len(values[s]) == 1:
        d2 = values[s]
        if not all(eliminate(values, s2, d2) for s2 in peers[s]):
            return False
    ## (2) If a unit u is reduced to only one place for a value d, then put it there.
    for u in units[s]:
        dplaces = [s for s in u if d in values[s]]
        if len(dplaces) == 0:
            return False  ## Contradiction: no place for this value
        # d can only be in one place in unit; assign it there
        elif len(dplaces) == 1 and not assign(values, dplaces[0], d):
            return False
    return values

def display(values):
    "Display these values as a 2-D grid."
    width = 1 + max(len(values[s]) for s in squares)
    line = "+".join(["-" * (width * 3)] * 3)
    for r in rows:
                values[r + c].center(width) + ("|" if c in "36" else "") for c in cols
        if r in "CF":

def solve(grid):
    return search(parse_grid(grid))

def some(seq):
    "Return some element of seq that is true."
    for e in seq:
        if e:
            return e
    return False

def search(values):
    "Using depth-first search and propagation, try all possible values."
    if values is False:
        return False  ## Failed earlier
    if all(len(values[s]) == 1 for s in squares):
        return values  ## Solved!
    ## Chose the unfilled square s with the fewest possibilities
    n, s = min((len(values[s]), s) for s in squares if len(values[s]) > 1)
    return some(search(assign(values.copy(), s, d)) for d in values[s])

def solve_all(grids, name="", showif=0.0):
    """Attempt to solve a sequence of grids. Report results.
    When showif is a number of seconds, display puzzles that take longer.
    When showif is None, don't display any puzzles."""

    def time_solve(grid):
        start = time.monotonic()
        values = solve(grid)
        t = time.monotonic() - start
        ## Display puzzles that take long enough
        if showif is not None and t > showif:
            if values:
            print("(%.5f seconds)\n" % t)
        return (t, solved(values))

    times, results = zip(*[time_solve(grid) for grid in grids])
    if (n := len(grids)) > 1:
            "Solved %d of %d %s puzzles (avg %.2f secs (%d Hz), max %.2f secs)."
            % (sum(results), n, name, sum(times) / n, n / sum(times), max(times))

def solved(values):
    "A puzzle is solved if each unit is a permutation of the digits 1 to 9."

    def unitsolved(unit):
        return {values[s] for s in unit} == set(digits)

    return values is not False and all(unitsolved(unit) for unit in unitlist)

def from_file(filename, sep="\n"):
    "Parse a file into a list of strings, separated by sep."
    return open(filename).read().strip().split(sep)  # noqa: SIM115

def random_puzzle(assignments=17):
    """Make a random puzzle with N or more assignments. Restart on contradictions.
    Note the resulting puzzle is not guaranteed to be solvable, but empirically
    about 99.8% of them are solvable. Some have multiple solutions."""
    values = {s: digits for s in squares}
    for s in shuffled(squares):
        if not assign(values, s, random.choice(values[s])):
        ds = [values[s] for s in squares if len(values[s]) == 1]
        if len(ds) >= assignments and len(set(ds)) >= 8:
            return "".join(values[s] if len(values[s]) == 1 else "." for s in squares)
    return random_puzzle(assignments)  ## Give up and make a new puzzle

def shuffled(seq):
    "Return a randomly shuffled copy of the input sequence."
    seq = list(seq)
    return seq

grid1 = (
grid2 = (
hard1 = (

if __name__ == "__main__":
    # solve_all(from_file("easy50.txt", '========'), "easy", None)
    # solve_all(from_file("top95.txt"), "hard", None)
    # solve_all(from_file("hardest.txt"), "hardest", None)
    solve_all([random_puzzle() for _ in range(99)], "random", 100.0)
    for puzzle in (grid1, grid2):  # , hard1):  # Takes 22 sec to solve on my M1 Mac.
        start = time.monotonic()
        t = time.monotonic() - start
        print("Solved: %.5f sec" % t)