[SUMMARY] Kalah (#58)

Both sides of the Kalah board are the same, with only the numbers of the slots
changing.  You can write a little code to make it so that your players need not
be concerned with this minor difference.  Here's an example from Adam Shelly:

	#Adapter class - rotates the board so that player's Kalah is always 6
	class KalahPlayer < Player
		def choose_move
		  n = (@side==KalahGame::TOP) ? 7 : 0
			@board = @game.board
			@board = @board.rotate n
			return get_move + n
		end
	  
		#simulate a move
		def simulate board,i
			b = board.dup
			stones,b[i]=b[i],0
			while stones > 0
				i = 0 if (i+=1) >12
				b[i]+=1
				stones-=1
			end
			if (0..5)===i and b[i]==1
				b[6]+= (b[i]+b[opposite(i)])
				b[i]=b[opposite(i)]=0
			end
			b
		end
		def opposite n
			12-n
		end
	  
	end

As the comment suggests, this is the Adapter Design Pattern at work.  Player's
original interface is that choose_move() should return a correct move regardless
of which side of the board it is playing from.  KalahPlayer changes that
interface to having get_move() see the board as if it was playing from the
bottom and return only moves in that range.

This change is accomplished by defining a choose_move() the rotates a board and
adds an offset to the players selected move, if needed.  When the player is on
the bottom, the board is rotated 0 and the selected move is added to an offset
of 0, resulting in no change either way.  When the player is on the top though,
the board the player sees is flipped 180 degrees (rotate(7)), and the player's
move is raised by 7 to place it on the proper side of the board.

This class also defines the helper method simulate(), which given a board and a
move, will return the resulting board.  This makes searching a lot easier for
the subclasses.

The above code makes use of a rotate() method we haven't seen.  It comes from a
few helpers added to Array:

	#Some helpers in Array
	class Array
		def rotate n
			a =dup
			n.times do a << a.shift end
			a
		end
		def sum
			inject(0){|s,e|s+=e}
		end
		#choose randomly between all items with given value
		def random_index value
			n=rand(find_all{|e|e==value}.size)
			each_with_index{|e,i| return i if e==value and (n-=1)<0 }
		end
	end

The first method is the rotate() call used earlier.  It's pretty simple, just
returning a copy of the Array with the elements cycled in a circular fashion the
requested number of times.

I really hope sum() needs no introduction by this point, as it's probably the
most popular helper method in Ruby Quiz solutions.  It just returns a summation
of all elements in the Array.

Finally, random_index() selects a random element equal to the passed in value. 
In other words, if an Array contained [1, 2, 3, 2] and you call random_index(2),
you have a 50/50 chance of getting 1 or 3 (the indices of the 2s) returned.

Now we're ready to actually build a player:

	#Tries to find the biggest increase in score for a turn
	class DeepScoreKalahPlayer < KalahPlayer
		def get_move
			best_move(@board)
		end
		def best_move board
			possible_scores = (0..5).map{|i| score_for(board,i)}
			possible_scores.index(possible_scores.max)
		end
	  
		#find the increase in score if we make move m
		def score_for board,m
			return -100 if board[m]<1         #flag invalid move
		  b, taketurn  = board,true
			while taketurn
				taketurn =  ((b[m]+m)%14 == 6)  #will we land in kalah?
				b = simulate b,m                               
				m = best_move(b) if taketurn             
			end
			b[6]-board[6]                     #how many points did we gain?
		end
	  
	end

This is a fairly uncomplicated player with the idea of find the most points I
can get with my move.  Remember, get_move() is the adapted interface, so that's
where we start.  All it does is call best_move() passing in the current board.

Now best_move() walks all possible moves calculating how many points we would
earn by making the move.  When it has that information, it selects the highest
point move and returns to index for that slot.

The last piece of this puzzle is score_for().  Given a board and a move, it
returns how many points you would gain by making the move.  This is slightly
complicated, because a move ending in the Kalah allows us to move again.  That
is handled with recursion by simply calling best_move() again on the new board. 
When that's all done, subtracting the Kalah we started with from the Kalah we
ended with gives us the points gained by the move or moves.

Adam doesn't quit there though, there's another player that inherits from this
one.  Let's take a look:

	#Tries to find the biggest increase in score for a turn
	#subtracts opponent's possible score
	class APessimisticKalahPlayer < DeepScoreKalahPlayer
		MaxDepth = 3
		def get_move
		  @level=0
			best_move(@board)
		end
		def best_move board
			return super(board) if (@level > MaxDepth)
			@level+=1
			possible_scores = (0..5).map{|i| 
				score_for(board,i) - worst_case(simulate(board,i)) 
			}
			@level-=1
			possible_scores.random_index(possible_scores.max)
		end
		#biggest score the opponent can get on this board
		def worst_case board
			worst = 0
			opp_board = board.rotate 7
			6.times {|i|
					s = score_for(opp_board, i)
					worst = s if worst < s
				}
				worst
		end
	end

This is a famous gaming algorithm known as a Minimax Search.  The idea is that
everyone is always going to make their best move, both you and your opponent. 
Given that, we want to find your best (or maximum) choice, but we also need to
consider the opponent's best response.  We want to chose the path that minimizes
the opponent's potential to get back at us.  The father ahead we can look in
this fashion, the more intelligent of a decision we can make.

The code above is very similar to the last player we examined.  The main new
element here is worst_case(), which is just a tool to find the opponent's best
move (worst for us).  Beyond that, you can see that get_move() and best_move()
have been reworked to search down to a constant depth.  In this case, a depth of
3 was chosen.  Of course we want to look as deep as possible, but memory and
execution time can make deeper searches unrealistic.

Note that best_move() now subtracts worst_case() from score_for(), so the
opponent's response move is weighted against our own.  This may even yield
negative values for moves, because the opponent has a response better than the
move we made.

If you want to take Minimax deeper, you have to cut down the work it has to do. 
A common technique is Alpha-Beta Pruning, which allows you to skip large
portions of the move tree in your search.  I'm not going to show the code here,
but be sure to look at David Balmain's code which walks father along this path.

My thanks to all the players who always share such interesting solutions with
us.

Tomorrow I have a real treat for all you Ruby Quiz fans, our first prize
awarding competition...