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The Ruby Programming Language by Yukihiro Matsumoto, David Flanagan

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A Sudoku Solver in Ruby

This chapter concludes with a nontrivial Ruby application to give you a better idea of what Ruby programs actually look like. We’ve chosen a Sudoku[*] solver as a good short to medium-length program that demonstrates a number of features of Ruby. Don’t expect to understand every detail of Example 1-1, but do read through the code; it is very thoroughly commented, and you should have little difficulty following along.

Example 1-1. A Sudoku solver in Ruby

# This module defines a Sudoku::Puzzle class to represent a 9x9
# Sudoku puzzle and also defines exception classes raised for 
# invalid input and over-constrained puzzles. This module also defines 
# the method Sudoku.solve to solve a puzzle. The solve method uses
# the Sudoku.scan method, which is also defined here.
# Use this module to solve Sudoku puzzles with code like this:
#  require 'sudoku'
#  puts Sudoku.solve(Sudoku::Puzzle.new(ARGF.readlines))
module Sudoku

  # The Sudoku::Puzzle class represents the state of a 9x9 Sudoku puzzle.
  # Some definitions and terminology used in this implementation: 
  # - Each element of a puzzle is called a "cell".
  # - Rows and columns are numbered from 0 to 8, and the coordinates [0,0]
  #   refer to the cell in the upper-left corner of the puzzle.
  # - The nine 3x3 subgrids are known as "boxes" and are also numbered from
  #   0 to 8, ordered from left to right and top to bottom. The box in
  #   the upper-left is box 0. The box in the upper-right is box 2. The
  #   box in the middle is box 4. The box in the lower-right is box 8.
  # Create a new puzzle with Sudoku::Puzzle.new, specifying the initial
  # state as a string or as an array of strings. The string(s) should use
  # the characters 1 through 9 for the given values, and '.' for cells
  # whose value is unspecified. Whitespace in the input is ignored.
  # Read and write access to individual cells of the puzzle is through the
  # [] and []= operators, which expect two-dimensional [row,column] indexing.
  # These methods use numbers (not characters) 0 to 9 for cell contents.
  # 0 represents an unknown value.
  # The has_duplicates? predicate returns true if the puzzle is invalid
  # because any row, column, or box includes the same digit twice.
  # The each_unknown method is an iterator that loops through the cells of
  # the puzzle and invokes the associated block once for each cell whose
  # value is unknown.
  # The possible method returns an array of integers in the range 1..9.
  # The elements of the array are the only values allowed in the specified
  # cell. If this array is empty, then the puzzle is over-specified and 
  # cannot be solved. If the array has only one element, then that element
  # must be the value for that cell of the puzzle.
  class Puzzle

    # These constants are used for translating between the external 
    # string representation of a puzzle and the internal representation.
    ASCII = ".123456789"
    BIN = "\000\001\002\003\004\005\006\007\010\011"

    # This is the initialization method for the class. It is automatically
    # invoked on new Puzzle instances created with Puzzle.new. Pass the input
    # puzzle as an array of lines or as a single string. Use ASCII digits 1
    # to 9 and use the '.' character for unknown cells. Whitespace, 
    # including newlines, will be stripped.
    def initialize(lines)
      if (lines.respond_to? :join)  # If argument looks like an array of lines
        s = lines.join              # Then join them into a single string
      else                          # Otherwise, assume we have a string
        s = lines.dup               # And make a private copy of it

      # Remove whitespace (including newlines) from the data
      # The '!' in gsub! indicates that this is a mutator method that
      # alters the string directly rather than making a copy.
      s.gsub!(/\s/, "")  # /\s/ is a Regexp that matches any whitespace

      # Raise an exception if the input is the wrong size.
      # Note that we use unless instead of if, and use it in modifier form.
      raise Invalid, "Grid is the wrong size" unless s.size == 81
      # Check for invalid characters, and save the location of the first.
      # Note that we assign and test the value assigned at the same time.
      if i = s.index(/[^123456789\.]/)
        # Include the invalid character in the error message.
        # Note the Ruby expression inside #{} in string literal.
        raise Invalid, "Illegal character #{s[i,1]} in puzzle"

      # The following two lines convert our string of ASCII characters
      # to an array of integers, using two powerful String methods.
      # The resulting array is stored in the instance variable @grid
      # The number 0 is used to represent an unknown value.
      s.tr!(ASCII, BIN)      # Translate ASCII characters into bytes
      @grid = s.unpack('c*') # Now unpack the bytes into an array of numbers

      # Make sure that the rows, columns, and boxes have no duplicates.
      raise Invalid, "Initial puzzle has duplicates" if has_duplicates?

    # Return the state of the puzzle as a string of 9 lines with 9 
    # characters (plus newline) each.  
    def to_s
      # This method is implemented with a single line of Ruby magic that
      # reverses the steps in the initialize() method. Writing dense code
      # like this is probably not good coding style, but it demonstrates
      # the power and expressiveness of the language.
      # Broken down, the line below works like this:
      # (0..8).collect invokes the code in curly braces 9 times--once
      # for each row--and collects the return value of that code into an
      # array. The code in curly braces takes a subarray of the grid
      # representing a single row and packs its numbers into a string.
      # The join() method joins the elements of the array into a single
      # string with newlines between them. Finally, the tr() method
      # translates the binary string representation into ASCII digits.
      (0..8).collect{|r| @grid[r*9,9].pack('c9')}.join("\n").tr(BIN,ASCII)

    # Return a duplicate of this Puzzle object.
    # This method overrides Object.dup to copy the @grid array.
    def dup
      copy = super       # Make a shallow copy by calling Object.dup
      @grid = @grid.dup  # Make a new copy of the internal data 
      copy               # Return the copied object

    # We override the array access operator to allow access to the 
    # individual cells of a puzzle. Puzzles are two-dimensional,
    # and must be indexed with row and column coordinates.
    def [](row, col)
      # Convert two-dimensional (row,col) coordinates into a one-dimensional
      # array index and get and return the cell value at that index
      @grid[row*9 + col]

    # This method allows the array access operator to be used on the 
    # lefthand side of an assignment operation. It sets the value of 
    # the cell at (row, col) to newvalue.
    def []=(row, col, newvalue)
      # Raise an exception unless the new value is in the range 0 to 9.
      unless (0..9).include? newvalue
        raise Invalid, "illegal cell value" 
      # Set the appropriate element of the internal array to the value.
      @grid[row*9 + col] = newvalue

    # This array maps from one-dimensional grid index to box number.
    # It is used in the method below. The name BoxOfIndex begins with a 
    # capital letter, so this is a constant. Also, the array has been
    # frozen, so it cannot be modified.
    BoxOfIndex = [

    # This method defines a custom looping construct (an "iterator") for
    # Sudoku puzzles.  For each cell whose value is unknown, this method
    # passes ("yields") the row number, column number, and box number to the 
    # block associated with this iterator.
    def each_unknown
      0.upto 8 do |row|             # For each row
        0.upto 8 do |col|           # For each column
          index = row*9+col         # Cell index for (row,col)
          next if @grid[index] != 0 # Move on if we know the cell's value 
          box = BoxOfIndex[index]   # Figure out the box for this cell
          yield row, col, box       # Invoke the associated block

    # Returns true if any row, column, or box has duplicates.
    # Otherwise returns false. Duplicates in rows, columns, or boxes are not
    # allowed in Sudoku, so a return value of true means an invalid puzzle.
    def has_duplicates?
      # uniq! returns nil if all the elements in an array are unique.
      # So if uniq! returns something then the board has duplicates.
      0.upto(8) {|row| return true if rowdigits(row).uniq! }
      0.upto(8) {|col| return true if coldigits(col).uniq! }
      0.upto(8) {|box| return true if boxdigits(box).uniq! }
      false  # If all the tests have passed, then the board has no duplicates

    # This array holds a set of all Sudoku digits. Used below.
    AllDigits = [1, 2, 3, 4, 5, 6, 7, 8, 9].freeze

    # Return an array of all values that could be placed in the cell 
    # at (row,col) without creating a duplicate in the row, column, or box.
    # Note that the + operator on arrays does concatenation but that the - 
    # operator performs a set difference operation.
    def possible(row, col, box)
      AllDigits - (rowdigits(row) + coldigits(col) + boxdigits(box))

    private  # All methods after this line are private to the class

    # Return an array of all known values in the specified row.
    def rowdigits(row)
      # Extract the subarray that represents the row and remove all zeros.
      # Array subtraction is set difference, with duplicate removal.
      @grid[row*9,9] - [0]

    # Return an array of all known values in the specified column.
    def coldigits(col)
      result = []                # Start with an empty array
      col.step(80, 9) {|i|       # Loop from col by nines up to 80
        v = @grid[i]             # Get value of cell at that index
        result << v if (v != 0)  # Add it to the array if non-zero
      result                     # Return the array

    # Map box number to the index of the upper-left corner of the box.
    BoxToIndex = [0, 3, 6, 27, 30, 33, 54, 57, 60].freeze

    # Return an array of all the known values in the specified box.
    def boxdigits(b)
      # Convert box number to index of upper-left corner of the box.
      i = BoxToIndex[b]
      # Return an array of values, with 0 elements removed.
        @grid[i],    @grid[i+1],  @grid[i+2],
        @grid[i+9],  @grid[i+10], @grid[i+11],
        @grid[i+18], @grid[i+19], @grid[i+20]
      ] - [0]
  end  # This is the end of the Puzzle class

  # An exception of this class indicates invalid input,
  class Invalid < StandardError

  # An exception of this class indicates that a puzzle is over-constrained
  # and that no solution is possible.
  class Impossible < StandardError

  # This method scans a Puzzle, looking for unknown cells that have only
  # a single possible value. If it finds any, it sets their value. Since
  # setting a cell alters the possible values for other cells, it 
  # continues scanning until it has scanned the entire puzzle without 
  # finding any cells whose value it can set.
  # This method returns three values. If it solves the puzzle, all three 
  # values are nil. Otherwise, the first two values returned are the row and
  # column of a cell whose value is still unknown. The third value is the
  # set of values possible at that row and column. This is a minimal set of
  # possible values: there is no unknown cell in the puzzle that has fewer
  # possible values. This complex return value enables a useful heuristic 
  # in the solve() method: that method can guess at values for cells where
  # the guess is most likely to be correct.
  # This method raises Impossible if it finds a cell for which there are
  # no possible values. This can happen if the puzzle is over-constrained,
  # or if the solve() method below has made an incorrect guess.
  # This method mutates the specified Puzzle object in place.
  # If has_duplicates? is false on entry, then it will be false on exit.
  def Sudoku.scan(puzzle)
    unchanged = false  # This is our loop variable

    # Loop until we've scanned the whole board without making a change.
    until unchanged 
      unchanged = true      # Assume no cells will be changed this time
      rmin,cmin,pmin = nil  # Track cell with minimal possible set
      min = 10              # More than the maximal number of possibilities

      # Loop through cells whose value is unknown.
      puzzle.each_unknown do |row, col, box|
        # Find the set of values that could go in this cell
        p = puzzle.possible(row, col, box)
        # Branch based on the size of the set p. 
        # We care about 3 cases: p.size==0, p.size==1, and p.size > 1.
        case p.size
        when 0  # No possible values means the puzzle is over-constrained
          raise Impossible
        when 1  # We've found a unique value, so set it in the grid
          puzzle[row,col] = p[0] # Set that position on the grid to the value
          unchanged = false      # Note that we've made a change
        else    # For any other number of possibilities
          # Keep track of the smallest set of possibilities.
          # But don't bother if we're going to repeat this loop.
          if unchanged && p.size < min
            min = p.size                    # Current smallest size
            rmin, cmin, pmin = row, col, p  # Note parallel assignment
    # Return the cell with the minimal set of possibilities.
    # Note multiple return values.
    return rmin, cmin, pmin

  # Solve a Sudoku puzzle using simple logic, if possible, but fall back
  # on brute-force when necessary. This is a recursive method. It either
  # returns a solution or raises an exception. The solution is returned
  # as a new Puzzle object with no unknown cells. This method does not 
  # modify the Puzzle it is passed. Note that this method cannot detect
  # an under-constrained puzzle.
  def Sudoku.solve(puzzle)
    # Make a private copy of the puzzle that we can modify.
    puzzle = puzzle.dup

    # Use logic to fill in as much of the puzzle as we can.
    # This method mutates the puzzle we give it, but always leaves it valid.
    # It returns a row, a column, and set of possible values at that cell.
    # Note parallel assignment of these return values to three variables.
    r,c,p = scan(puzzle)

    # If we solved it with logic, return the solved puzzle.
    return puzzle if r == nil
    # Otherwise, try each of the values in p for cell [r,c].
    # Since we're picking from a set of possible values, the guess leaves
    # the puzzle in a valid state. The guess will either lead to a solution
    # or to an impossible puzzle. We'll know we have an impossible
    # puzzle if a recursive call to scan throws an exception. If this happens
    # we need to try another guess, or re-raise an exception if we've tried
    # all the options we've got.
    p.each do |guess|        # For each value in the set of possible values
      puzzle[r,c] = guess    # Guess the value
        # Now try (recursively) to solve the modified puzzle.
        # This recursive invocation will call scan() again to apply logic
        # to the modified board, and will then guess another cell if needed.
        # Remember that solve() will either return a valid solution or 
        # raise an exception.  
        return solve(puzzle)  # If it returns, we just return the solution
      rescue Impossible
        next                  # If it raises an exception, try the next guess

    # If we get here, then none of our guesses worked out
    # so we must have guessed wrong sometime earlier.
    raise Impossible

Example 1-1 is 345 lines long. Because the example was written for this introductory chapter, it has particularly verbose comments. Strip away the comments and the blank lines and you’re left with just 129 lines of code, which is pretty good for an object-oriented Sudoku solver that does not rely on a simple brute-force algorithm. We hope that this example demonstrates the power and expressiveness of Ruby.

[*] Sudoku is a logic puzzle that takes the form of a 9×9 grid of numbers and blank squares. The task is to fill each blank with a digit 1 to 9 so that no row or column or 3×3 subgrid includes the same digit twice. Sudoku has been popular in Japan for some time, but it gained sudden popularity in the English-speaking world in 2004 and 2005. If you are unfamiliar with Sudoku, try reading the Wikipedia entry (http://en.wikipedia.org/wiki/Sudoku) and try an online puzzle (http://websudoku.com/).

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