[SUMMARY] Chess960 (#106)

There are a surprising number of ways to think about this problem, each leading
to a different solution.  Let's examine several.

First, you can think of this as a constraints problem.  We have the rules of a
board setup which are the constraints and any board meeting those rules is one
of the 960 positions.  Way back when we did the Constraint Processing Ruby Quiz,
I learned that the Amb library is a fun way to handle such problems.  Here's my
solution, using Amb:

	require "amb"
	
	setup = Amb.new
	count = 0
	seen  = Hash.new
	begin
	  squares = Array.new(8) { setup.choose(*%w[r n b q k b n r]) }
	  
	  %w[r n b].each do |piece|
	    setup.assert(squares.select { |s| s == piece }.size == 2)
	  end
	  %w[k q].each do |piece|
	    setup.assert(squares.select { |s| s == piece }.size == 1)
	  end
	  king = squares.index("k")
	  setup.assert(squares.index("r") < king)
	  setup.assert(squares.rindex("r") > king)
	  setup.assert((squares.index("b") + squares.rindex("b")) % 2 == 1)
	  board = squares.join(' ')
	  setup.assert(seen[board].nil?)
	  
	  puts "#{count += 1}: #{board}"
	  
	  seen[board] = true
	  setup.failure
	rescue
	  # do nothing, we're done
	end

What I really love above this approach is that it requires almost no thought. 
All I am doing here is translating the rules to code.  Ruby and Amb do the rest.

The above code works by building an Amb object that is considering piece
placement in eight squares.  After that, I define the rules for placement.

Since I gave it all eight pieces as possible choices for each square, the first
two rules have to establish the allowed counts for each piece.  This prevents
Amb from building a position of eight rooks or similar wrong setups.

For the next two rules I locate the king and verify that we have a rook to the
left of him as well as to the right.  The following rule adds the locations of
the two bishops and verifies that we got an odd number.  This ensures the
bishops are on different colors, since an even plus an odd will be odd but even
plus even or odd plus odd both yield even numbers.  This group of rules covers
the constraints from the quiz description.

The final rule prevents duplicate positions being found.  It's needed because
positions like the following example seem different to the computer, but not to
a chess player:

	N1 B1 B2 R1 K R2 Q N2
	N2 B2 B1 R2 K R1 Q N1

With the rules in place, Amb will have found a viable solution.  I print that
out, then manually trigger a failure(), to cause it to find another.  When it
fails to find one an Exception will be thrown.  I catch that and exit quietly
since we will have seen all 960 positions at that point.

The downside of this approach is that Amb uses Ruby's continuations to under the
hood to backtrack and find new solutions.  Those are a bit on the slow side and
this simple script of mine takes about six and a half minutes to complete, on my
box.  One solution to this is just to generate the positions once and cache them
for future runs but there were other solutions that could think a lot faster.

For a faster solution, let's shift how we are thinking about the problem. 
Another approach is to think of this challenge as a permutations problem.  We
really just need to examine all possible permutations of the eight pieces and
select those that match the quiz rules.  Multiple solutions did this, including
the following code from David Tran:

	def permutation(pieces)
	 return [pieces] if pieces.length <= 1
	 result = []
	 pieces.uniq.each do |p|
	   _pieces = pieces.dup
	   _pieces.delete_at(pieces.index(p))
	   permutation(_pieces).each do |perm|
	     result << (perm << p)
	   end
	 end
	 result
	end
	
	results = permutation("RNBKQBNR".split(//)).select do |position|
	 r1 = position.index('R')
	 r2 = position.rindex('R')
	 b1 = position.index('B')
	 b2 = position.rindex('B')
	 k = position.index('K')
	 r1 < k && k < r2 && ((b1+b2) % 2 != 0)
	end
	
	puts "Total positions = #{results.length}"
	puts results[rand(results.length)].join(' ')

The permutation() method here is a recursive generator of all possible
permutations.  It works by adding each() uniq() piece to all possible smaller
permutations.  Those are found by removing one piece from the set each time and
recursing.

The rest of the code calls that method and then select()s the positions matching
the quiz rules.  Those rules are almost identical to my implementation of them
that we examined earlier.

This code does the same thing as mine but runs in under a second on my box.

Shifting the approach again, several methods have been developed to help players
create positions using these rules as needed, some using dice.  Bodlaender's
dice-rolling method is a pretty easy system to translate into code and Jamie
Macey did just that:

	class Chess960
	 attr_reader :board_id, :board
	
	 def initialize
	   @board = generate_board(bodlaender_line)
	 end
	
	 def generate_board(white)
	   # Black's starting line is mirror of white's
	   black = white.map{|piece| piece.downcase}
	
	   # middle of board is always the same
	   middle = [
	     %w(p p p p p p p p),
	     %w(_ _ _ _ _ _ _ _),
	     %w(_ _ _ _ _ _ _ _),
	     %w(_ _ _ _ _ _ _ _),
	     %w(_ _ _ _ _ _ _ _),
	     %w(P P P P P P P P)
	   ]
	
	   # add back rows
	   [black] + middle + [white]
	 end
	
	 def bodlaender_line
	   free = (0...8).to_a
	   white = []
	
	   dark_bishop = rand(4) * 2
	   light_bishop = rand(4) * 2 + 1
	   white[dark_bishop] = 'B'
	   white[light_bishop] = 'B'
	   free.delete(dark_bishop)
	   free.delete(light_bishop)
	
	   queen = rand(6)
	   white[free[queen]] = 'Q'
	   free.delete_at(queen)
	
	   knight1 = rand(5)
	   white[free[knight1]] = 'N'
	   free.delete_at(knight1)
	   knight2 = rand(4)
	   white[free[knight2]] = 'N'
	   free.delete_at(knight2)
	
	   white[free[0]] = 'R'
	   white[free[1]] = 'K'
	   white[free[2]] = 'R'
	   white
	 end
	end

The first two methods are trivial with initialize() kicking off the process and
generate_board() building a board representation.  The interesting stuff happens
in bodlaender_line().

This algorithm works with a collection of free indices and an Array of the final
piece arrangement.  As pieces are placed in the Array, those indices are pulled
from the free list so they won't be reused.

The first step is to place both bishops.  That's done by choosing a random
number between one and four and placing it on the selected light or dark square.
After that, a random selection places the queen on one of the six remaining
squares.  The same technique is used to place the knights on the remaining five
and then four squares.  That leaves us three empty squares which must be R, K,
and R, in that order, to satisfy the rules.

Making one more leap of thought with regard to this problem, there has been an
effort to enumerate all 960 positions.  This has a lot of value to chess players
since we can record the game using our usual techniques and just add a note
like, "Started from Chess960 position #351."  Even better, once you have a
system for enumerating the positions, you can use that system to build the
entire list or select a position.  Morton Goldberg gave us the code for that:

	class Chess960
	   BISHOP_TABLE = [
	      "BB------", #0
	      "B--B----", #1
	      "B----B--", #2
	      "B------B", #3
	      "-BB-----", #4
	      "--BB----", #5
	      "--B--B--", #6
	      "--B----B", #7
	      "-B--B---", #8
	      "---BB---", #9
	      "----BB--", #10
	      "----B--B", #11
	      "-B----B-", #12
	      "---B--B-", #13
	      "-----BB-", #14
	      "------BB"  #15
	   ]
	
	   N5N_TABLE = [
	      "NN---", #0
	      "N-N--", #1
	      "N--N-", #2
	      "N---N", #3
	      "-NN--", #4
	      "-N-N-", #5
	      "-N--N", #6
	      "--NN-", #7
	      "--N-N", #8
	      "---NN"  #9
	   ]
	
	   # ...

The official numbering scheme, invented by Reinhard Scharnagl, works by using
simple charts to position the pieces.  Above you can see Morton's translation of
the two charts he will use, giving piece placements and their indices.  The
knight charts are narrower because they are placed after the bishops and queen,
taking three squares out of consideration.

Here's the main algorithm from Morton's code (with a minor fix from me):

	   # ...
	   
	   def initialize(number)
	      q, @bishop_index = number.divmod 16
	      @knight_index, @queen_index = q.divmod 6
	      @white_pieces = BISHOP_TABLE[@bishop_index].split('')
	      @white_pieces[nth_dash(@queen_index)] = 'Q'
	      knights = N5N_TABLE[@knight_index]
	      m = knights.index('N')
	      n = knights.index('N', m + 1)
	      m, n = nth_dash(m), nth_dash(n)
	      @white_pieces[m] = 'N'
	      @white_pieces[n] = 'N'
	      @white_pieces[@white_pieces.index('-')] = 'R'
	      @white_pieces[@white_pieces.index('-')] = 'K'
	      @white_pieces[@white_pieces.index('-')] = 'R'
	   end
	   
	   def nth_dash(n)
	      dashes = []
	      @white_pieces.each_with_index { |ch, i| dashes << i if ch == '-' }
	      dashes[n]
	   end
	   
	   # ...

Most of the clever work is done in initialize().  First, a little math is used
to find the lookup index on the bishop's chart, an index for the queen, and an
index on the knight's chart.  The row selected from the bishop's chart becomes
the basis for the final arrangement of pieces and nth_dash() is used to properly
slot the queen in that template.  The knight's are then pulled from their chart
by index and nth_dash() is again used to place them.  The three squares left
then must be a rook, king, and rook in that order.

The rest of Morton's code is just for displaying the results:

	   # ...
	   
	   def inspect
	      @white_pieces.join
	   end
	
	   def to_s
	      white_pieces = @white_pieces.join + "\n"
	      white_pawns = 'P' * 8 + "\n"
	      black_pieces = white_pieces.downcase
	      black_pawns = 'p' * 8 + "\n"
	      empty = ('.' * 8 + "\n") * 4
	      black_pieces + black_pawns + empty + white_pawns + white_pieces
	   end
	end
	
	if __FILE__ == $0
	   begin
	      if ARGV.empty? then n = 1 + rand(960)
	      else
	         n = ARGV.first.to_i
	         raise StandardError unless (1..960).include?(n)
	      end
	      puts "Initial position #{n}"
	      print Chess960.new(n).to_s
	   rescue StandardError
	      puts "Usage:  #{$PROGRAM_NAME} [<integer>]"
	      puts "where <integer> is in 1..960"
	      puts "Omitting <integer> produces a random initial position"
	   end
	end

If you want to read more about the systems for assigning pieces, check out:

	http://en.wikipedia.org/wiki/Chess960_starting_position

and:

	http://en.wikipedia.org/wiki/Chess960_Enumbering_Scheme

There were a lot more creative elements in the solutions I didn't cover.  Jamie
Macey even built a complete Camping application to display the positions. 
Definitely take the time to look over them.  It's worth it.

My thanks to all for another great quiz.  I'm a big chess nut some problems like
this always thrill me.

There will be no Ruby Quiz tomorrow.  I'll be busy having a merry Christmas and
I wish the same for others.  See you in a week!