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Now, let s turn to the recursive part of the solution. When you have your base case covered, the recursive case may correctly assume (by induction) that all results from lower levels (the queens with higher numbers) are correct. So what you have to do is add an else clause to the if statement in the previous implementation of the queens function.
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What results do you expect from the recursive call You want the positions of all the lower queens, right Let s say they are returned as a tuple. In that case you probably have to change your base case to return a tuple as well (of length one) but I get to that later. So, you re supplied with one tuple of positions from above, and for each legal position of the current queen, you are supplied with a tuple of positions from below. All you have to do to keep things flowing is to yield the result from below with your own position added to the front: ... else: for pos in range(num): if not conflict(state, pos): for result in queens(num, state + (pos,)): yield (pos,) + result The for pos and if not conflict parts of this are identical to what you had before so you can rewrite this a bit to simplify the code. Let s add some default arguments as well: def queens(num=8, state=()): for pos in range(num): if not conflict(state, pos): if len(state) == num-1: yield (pos,) else: for result in queens(num, state + (pos,)): yield (pos,) + result If you find the code hard to understand, you might find it helpful to formulate what it does in your own words. (And, you do remember that the comma in (pos,) is necessary to make it a tuple, and not simply a parenthesized value ) The queens generator gives you all the solutions (that is, all the legal ways of placing the queens): >>> list(queens(3)) [] >>> list(queens(4)) [(1, 3, 0, 2), (2, 0, 3, 1)] >>> for solution in queens(8): ... print solution ... (0, 4, 7, 5, 2, 6, 1, 3) (0, 5, 7, 2, 6, 3, 1, 4) ... (7, 2, 0, 5, 1, 4, 6, 3) (7, 3, 0, 2, 5, 1, 6, 4) >>> If you run queens with eight queens, you see a lot of solutions flashing by. Let s find out how many:
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>>> len(list(queens(8))) 92
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Wrapping It Up
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Before leaving the queens, let s make the output a bit more understandable. Clear output is always a good thing because it makes it easier to spot bugs, among other things. def prettyprint(solution): def line(pos, length=len(solution)): return '. ' * (pos) + 'X ' + '. ' * (length-pos-1) for pos in solution: print line(pos) Note that I ve made a little helper function inside prettyprint. I put it there because I assumed I wouldn t need it anywhere outside. In the following, I print out a random solution to satisfy myself that it is correct: >>> >>> . . . X . . X . . . . . . . . . import random prettyprint(random.choice(list(queens(8)))) . . . X . . . . . . . . . . . . X . . . . . . . . X . . . . . . . . . X . . X . . . X . . . . . This drawing corresponds to the diagram in Figure 9-2. Fun to play with Python, isn t it
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Figure 9-2. One of many possible solutions to the Eight Queens problem
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A Quick Summary
A lot of magic here. Let s take stock: New-style classes. The way classes work in Python is changing. In Python 2.2, new-style classes were introduced, and they provide several new features (for example, they work with super and property, while old-style classes do not). To create a new-style class you must subclass object, either directly or indirectly, or set the __metaclass__ property. Magic methods. Several special methods (with names beginning and ending with double underscores) exist in Python. These methods differ quite a bit in function, but most of them are called automatically by Python under certain circumstances. (For example, __init__ is called after object creation.) Constructors. These are common to many object-oriented languages, and you ll probably implement one for almost every class you write. Constructors are named __init__ and are automatically called right after an object is created. Overriding. A class can override methods (or any other attributes) defined in its superclasses simply by implementing the methods. If the new method needs to call the overridden version, it can either call the unbound version from the superclass directly (old-style classes) or use the super function (new-style classes). Sequences and mappings. Creating a sequence or mapping of your own requires implementing all the methods of the sequence and mapping protocols, including such magic methods as __getitem__ and __setitem__. By subclassing list (or User List) and dict (or UserDict) you can save a lot of work. Iterators. An iterator is simply an object that has a next method. Iterators can be used to iterate over a set of values. When there are no more values, the next method should raise a StopIteration exception. Iterable objects have an __iter__ method, which returns an iterator, and can be used in for loops, just like sequences. Often, an iterator is also iterable, that is, it has an __iter__ method returning the iterator itself. Generators. A generator function (or method) is a function (or method) that contains the keyword yield. When called, the generator function returns a generator, which is a special type of iterator. Eight Queens. The Eight Queens problem is well known in computer science and lends itself easily to implementation with generators. The goal is to position eight queens on a chess board so that none of the queens is in a position from which she can attack any of the others.
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