[SUMMARY] Magic Fingers (#120)

Eric I. said it best when he said the real trick of this quiz is figuring out a
way to convince someone of the outcome beyond the shadow of a doubt.  Eric's own
code convinced me, after we clarified my mistakes in the rules.

Eric's code outputs a table of your moves compared with the opponent's moves. 
At each step, it tells you the move to make based on the following priorities:

	1. It suggests a forced win if it can find one
	2. It aims for draw if there is no forced win
	3. As a last resort, it will stall a loss as long as possible to increase
	   the chances of your opponent making an error

The first one is really the point of interest for this quiz.  There's just not
that many different hand positions in this game and a sufficiently deep search
can find the wins.  Here's the chart Eric's code shows for the game:

	       01   02   03   04   11   12   13   14   22   23   24   33   34   44
	     ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
	01:  -1T1 -1T2 -1T3 +1T4 -1T1 -1T2 -1T1 +1T4 -1T2 -1T2 -1T4 -1T3 -1T4 -1T4
	02:  +C11 -C11 +2T3 +2T4 +C11 -C11 +2T3 +2T4 -C11 +2T3 +2T4 -C11 -C11 -C11
	03:  +C21 +3T2 +3T3 +3T4 -C21 +3T2 +3T3 +3T4 -C21 -C21 -C21 -C21 -C21 -C21
	04:  +4T1 +4T2 +4T3 +4T4 +C31 +C22  C22 +C22  C22 +C22  C22 -C22 -C22 -C22
	11:  +C02 +1T2 -1T3 +1T4 -1T1 +C02 -1T3 +1T4  C02 -1T2 -1T4 -1T3 +1T4 -1T4
	12:  +C03 -2T2 +2T3 +2T4 +C03 +2T2 +2T3 +2T4 -1T2 +2T3 +2T4 -1T3 -2T3 -1T4
	13:  +C22 +3T2 +3T3 +3T4 +3T1 +C22 +3T3 +1T4  C22 +C22  C22 -C22  3T3  1T4
	14:  +4T1 +4T2 +4T3 +4T4 +C32 -C32 -C32 +C32 -C32 -4T3 -4T2 -1T3 -4T3 -1T4
	22:  +C13  2T2 +2T3 +2T4 +C13 +2T2 +2T3 +2T4  2T2 +2T3 +2T4 +2T3 +2T4  2T4
	23:  +2T1 +3T2 +3T3 +3T4 +3T1 +3T2 +3T3 +3T4 -3T2 +3T2 -3T2 +3T3 +3T4 -2T4
	24:  +4T1 +4T2 +2T3 +2T4 +4T1 +4T2 +2T3 +2T4  2T2 +4T2  4T2 +4T3 +4T4  2T4
	33:  +C24 +3T2 +3T3 +3T4 +3T1 +3T2 +3T3 +3T4 +3T2 +3T2 +3T4 +3T3 +3T4 +3T4
	34:  +4T1 +4T2 +4T3 +4T4 +4T1 +4T2 +4T3 +4T4 +4T2 +4T3 +4T4 +4T3 +4T4 +4T4
	44:  +4T1 +4T2 +4T3 +4T4 +4T1 +4T2 +4T3 +4T4 +4T2 +4T3 +4T4 +4T3 +4T4 +4T4

Columns and rows are labeled in normalized hand positions, meaning that a left
hand of one finger and a right hand of two fingers is the same as a left of two
with a right of one.

You find the row for your hands and cross reference it with the column of the
opponent's hands.  The cell where the two meet lists your best move, be it a
touch or a clap.  Don't worry too much about the format of those moves, but know
this:  a plus indicates that you have a forced win and a minus indicates that
you face a forced loss.

Using that knowledge you can look up the starting position at row 11, column 11.
You will find a minus there telling you that your best move still leads to a
forced loss.  That's correct.  The second player can always win a game of Magic
Fingers.

Now let's examine how the code reaches this conclusion.  First, let's look at
the class used to represent the current game state:

	class GameState
	  attr_reader :hands
	
	  def initialize(hands = [[1, 1], [1, 1]])
	    @hands = hands
	  end
	
	  def do_turn(move)
	    new_hands, description1, description2 =
	      *move.call(@hands[0].dup, @hands[1].dup)
	    [GameState.new([new_hands[1], new_hands[0]]),
	      description1,
	      description2]
	  end
	
	  def to_s
	    result = ""
	    @hands.each_index do |i|
	      result << "#{i+1}: "
	      result << '-' * (5 - @hands[i][0])
	      result << '|' * @hands[i][0]
	      result << ' '
	      result << '|' * @hands[i][1]
	      result << '-' * (5 - @hands[i][1])
	      result << "\n"
	    end
	    result
	  end
	
	  def game_over?
	    @hands[0][0] == 0 && @hands[0][1] == 0 ||
	      @hands[1][0] == 0 && @hands[1][1] == 0
	  end
	
	  def score
	    if @hands[0][0] == 0 && @hands[0][1] == 0 : -1
	    elsif @hands[1][0] == 0 && @hands[1][1] == 0 : 1
	    else 0
	    end
	  end
	
	  def eql?(other)
	    @hands == other.hands
	  end
	
	  def hash
	    @hands[0][0] + 5 * @hands[0][1] + 25 * @hands[1][0] +
	      125 * @hands[1][1]
	  end
	end
	
	# ...

Most of this class is very straightforward.  A GameState is created by providing
the current values of the two hands.  You can then query the resulting object
for a win, loss, or yet unknown score(), whether or not it is game_over?(), or a
String representation of the hands.  The class also defines eql?() and hash() in
terms of the hand counts so these objects can be used as keys in a Hash.

The only semi-involved method is do_turn(), which takes a Proc that will perform
a move and uses that to create the resulting GameState object.  You will see how
this method is used when we get a little farther.

Next we will look into generating possible moves:

	# ...
	
	HandNames        = ["left hand", "right hand"]
	AllowClapsToZero = false
	
	def generate_touches
	  result = []
	  (0..1).each do |from_hand|
	    (0..1).each do |to_hand|
	      result << Proc.new do |player_hands, opponent_hands|
	        raise "cannot touch from empty hand" if
	          player_hands[from_hand] == 0
	        raise "cannot touch to empty hand" if
	          opponent_hands[to_hand] == 0
	        description1 =
	          "touches #{HandNames[from_hand]} to opponent's " +
	          "#{HandNames[to_hand]}"
	        description2 = "#{player_hands[from_hand]}T" +
	                       "#{opponent_hands[to_hand]}"
	        opponent_hands[to_hand] += player_hands[from_hand]
	        opponent_hands[to_hand] = 0 if opponent_hands[to_hand] >= 5
	        [[player_hands, opponent_hands], description1, description2]
	      end
	    end
	  end
	  result
	end
	
	def generate_claps
	  result = []
	  (0..1).each do |from_hand|
	    to_hand = 1 - from_hand
	    (1..4).each do |fingers|
	      result << Proc.new do |player_hands, opponent_hands|
	        raise "do not have enough fingers on " +
	              "#{HandNames[from_hand]}" unless
	          player_hands[from_hand] >= fingers
	        raise "#{HandNames[to_hand]} would end up with five or more " +
	              "fingers" if
	          !AllowClapsToZero && player_hands[to_hand] + fingers >= 5
	        raise "cannot end up with same number combination after clap" if
	          player_hands[from_hand] - fingers == player_hands[to_hand]
	        description1 = "claps to transfer #{fingers} fingers from " +
	          "#{HandNames[from_hand]} to #{HandNames[to_hand]}"
	        player_hands[from_hand] -= fingers
	        player_hands[to_hand] += fingers
	        player_hands[to_hand] = 0 if player_hands[to_hand] >= 5
	        description2 = "C#{player_hands[from_hand]}"+
	                       "#{player_hands[to_hand]}"
	        [[player_hands, opponent_hands], description1, description2]
	      end
	    end
	  end
	  result
	end
	
	Moves = generate_claps + generate_touches
	
	# ...

The main work here is in the two almost identical methods.  They work by filling
an Array with Proc objects that generate touch and clap moves when passed a pair
of hands.  These Procs return the new hands and descriptions of the move that
are used in the chart we saw earlier.  They also include a good deal of error
handling to prevent illegal moves, as you can see.  Finally the Moves constant
is populated with the results of a call to each.

This next method is the work horse of this solution.  This is a recursive depth
first search of the available moves.  It limits the recursion, to keep things
like draws from creating infinite loops, and memoizes GameState objects to keep
from redoing work.  Let's take it in slices:

	# ...
	
	Levels = 25
	Memo   = Hash.new
	
	def pick_move(state, levels = Levels)
	  return [state.score, nil, nil, nil] if levels <= 0 || state.game_over?
	
	  memoed_move = Memo[state]
	  if memoed_move && memoed_move[0] >= levels
	    # use memoed values if levels used meets or exceeds my levels
	    best_move = memoed_move[1]
	    best_score = memoed_move[2]
	  else
	    # ...

Right off the bat, we see the code that controls when recursion stops.  As soon
as the recursion limit is reached or a game is over, the code tosses a final
score back up the stack.

When that's not the case, the method moves into search mode.  The first step is
to check for a memoized answer that would short circuit the need to search at
all.  When we don't have that for the current position though, it's time to
recurse:

	    # ...
	    
	    # otherwise, calculate values recursively
	    best_score = nil
	    best_move = nil
	
	    # try each of the possible moves on this state and generate an
	    # array of the results of those choices
	    move_choices = Moves.map do |move|
	      begin
	        # determine the new state if the chosen move is applied
	        new_state, description1, description2 = *state.do_turn(move)
	
	        # recursively determine the score for this move (i.e., this
	        # state); negate the score returned since it's in terms of
	        # opponent (i.e., a win for them is a loss for us)
	        score = -pick_move(new_state, levels - 1)[0]
	
	        # increment score (by shifting away from zero) in order to be
	        # able to treat is as a count of the number of moves to a win
	        # or a loss
	        score += score / score.abs unless score.zero?
	
	        [score, move, description1, description2]
	      rescue Exception => e
	        nil  # the move was ilegal
	      end
	    end
	    
	    # ...

Here we see the Array of Proc move generators and the do_turn() method of
GameState come together.  Each Proc is passed in one-at-a-time to generate the
resulting GameState.  Remember that those Procs toss Exceptions whenever an
illegal move is found and this code uses a rescue clause to skip over such
moves.  The new state is then recursed into by pick_move() to fetch a resulting
score.  That score will be from the opponent's point of view, so it has to be
negated to count for our point of view.

When we have the moves, it's time to hunt for winners:

	    # ...
	    
	    # remove nils that were generated by illegal moves
	    move_choices = move_choices.select { |option| option }
	
	    # select and sort only those with positive (i.e., winning scores)
	    winning_choices = move_choices.
	      select { |option| option[0] > 0 }.
	      sort_by { |option| option[0] }
	
	    unless winning_choices.empty?
	      # if there's a winning option, choose the one that leads to a
	      # with the least number of moves
	      selected = winning_choices.first
	    else
	      # otherwise, choose a move that leads to a tie (preferable) or a
	      # loss but in the greatest number of moves (to increase
	      # opponent's opportunities to make a mistake)
	      move_choices = move_choices.sort_by { |option| option[0] }
	      if move_choices.last[0] == 0
	        selected = move_choices.last
	      else
	        selected = move_choices.first
	      end
	    end
	
	    best_score = selected[0]
	    best_move = selected[1..3]
	
	    # store the best move determined for future use
	    Memo[state] = [levels, best_move, best_score]
	  end
	
	  [best_score] + best_move
	end
	
	# ...

The first line is just a long-hand way to write move_choices.compact! and the
second line filters the legal moves down to winning moves.  If we have winning
moves, the quickest kill is selected.  Otherwise, the code checks draws and
losses as I described earlier.  At this point we finally know a best move and
score.  Those are memoized for future calls and passed back up the stack to the
calling code.

The next step is to put this method to work by handing it all possible hand
combinations:

	# ...
	
	# Returns a string indicating win or loss depending on score.
	def score_symbol(score)
	  if score > 0 : '+'
	  elsif score < 0 : '-'
	  else ' '
	  end
	end
	
	
	# Calculate the best move given every finger combination, and store in
	# the results hash.
	results = Hash.new
	1.upto(4) do |left1|
	  0.upto(left1) do |right1|
	    key1 = "#{right1}#{left1}"
	    results[key1] = Hash.new
	    1.upto(4) do |left2|
	      0.upto(left2) do |right2|
	        state = GameState.new([[left1, right1], [left2, right2]])
	        score, move, description1, description2 = *pick_move(state, 40)
	        key2 = "#{right2}#{left2}"
	        results[key1][key2] = score_symbol(score) + description2
	      end
	    end
	  end
	end
	
	# ...

This is just a brute force generation of positions, their scores, and the best
moves to make in them.  Everything shown in Eric's output is built here using
the tools we have been examining.

Speaking of that chart, drawing that is our last bit of code:

	# ...
	
	# display instructions
	puts <<EOS
	INSTRUCTIONS
	
	If it's your turn, select the row that describes your two hands.  Then
	select the column that describes your opponent's two hands.  The cell
	at the intersection will tell you how to move and what to expect.
	
	A leading "+" indicates there is a guaranteed way to win.  A leading
	"-" tells you that if the opponent plays perfectly, you will lose.  If
	neither of those symbols is present, then if you and your opponent
	play well, neither of you will ever win.
	
	The rest of the cell tells you what type of move to make.  A "T"
	represents a touching move, telling you which finger of yours first to
	user first, and which finger of the opponent to touch.  A "C"
	represents a clapping move, and it tells you the finger counts should
	end up with after the clap.
	
	EOS
	
	
	# display move strategy table
	line1 = "    " + results.keys.sort.map { |key1| "   #{key1}" }.join
	puts line1
	puts line1.gsub(/\ \ \d\d/, '----')
	results.keys.sort.each do |key1|
	  print "#{key1}: ",
	    results[key1].keys.sort.
	                       map { |key2| " #{results[key1] [key2]}" }.join,
	    "\n"
	end

This code is just boring print calls to display the chart.  There shouldn't be
anything surprising here.

My thanks to the few brave souls that toughed out my surprisingly challenging
problem.  I didn't realize it would be so much work, but you guys made it look
easy enough, as usual.

Tomorrow we have a super easy code breaking problem so I want to see all you
beginners playing!