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Solving the Knight's Tour Puzzle In 60 Lines of Python
ttsiod writes "When I was a kid, I used to play the Knight's Tour puzzle with pen and paper: you simply had to pass once from every square of a chess board, moving like a Knight. Nowadays, I no longer play chess; but somehow I remembered this nice little puzzle and coded a 60-line Python solver that can tackle even 100x100 boards in less than a second. Try beating this, fellow coders!"
There. I did it in one line of code.
#!/usr/bin/env python import sys g_sqSize = -1 # the board size, passed at runtime g_board = [] # the board will be constructed as a list of lists def main(): global g_sqSize if len(sys.argv) != 2: g_sqSize = 8 # Default: Fill the normal 8x8 chess board else: try: g_sqSize = int(sys.argv[1]) # or, the NxN the user wants except: print "Usage: " + sys.argv[0] + " " sys.exit(1) for i in xrange(0, g_sqSize): g_board.append(g_sqSize*[0]) # Fill the board with zeroes Fill(0,0,1) # Start the recursion with a 1 in the upper left print "No solution found" # if the recursion returns, it failed def InRangeAndEmpty(ty,tx): # check if coordinates are within the board return ty>=0 and tx>=0 and ty
... here: http://www.tri.org.au/tourcode.html
wrapper(Size, [X, Y], Path) :- :- :- :-
X == 1,
Y == 1,
Depth is Size * Size - 1,
worker(Size, [X, Y], Depth, [], ReversedPath),
reverse(ReversedPath, Path),
write(Path), nl.
worker(_, State, 0, CurrentPath, [State|CurrentPath]).
worker(Size, State, Depth, CurrentPath, FinalPath)
DepthM1 is Depth - 1,
move_generator(Size, State, NewState),
not(checker(NewState, CurrentPath)),
worker(Size, NewState, DepthM1, [State|CurrentPath], FinalPath).
checker(State, [State|_]).
checker(State, [_|StateList])
checker(State, StateList).
move_generator(Size, [X, Y], [NewX, NewY])
move(MoveX, MoveY),
NewX is X + MoveX, NewX == 1,
NewY is Y + MoveY, NewY == 1.
move(1, 2).
move(2, 1).
move(2, -1).
move(1, -2).
move(-1, -2).
move(-2, -1).
move(-2, 1).
move(-1, 2).
Except for the print statement....
#!/usr/bin/perl
use Chess;
$knight = Chess::Piece::Knight->new();
$board = Chess::Board->new(100, 100, setup => {
$knight => "a1";
});
$knight->tour()->show();
With the "added intelligence" of the second version, the recursive search devolved into a linear one since the very first attempt at each step will lead to a good solution (add a print to the backtracking part and see if this isn't the case).
So you might as well convert the recursion into a loop and eliminate the stack overflows for large boards.
It's even less readable that PERL. Shit, I didn't think it's possible. And I used to "program" in PERL...
"an experienced, industrious, ambitious, and often, quite often, picturesque liar" - Mark Twain
Here's a solution in 14 lines of APL. I'm pretty sure they could've made it shorter, but readability would've been even worse. APL has been called a "write-only language".
-- "At Microsoft, quality is job 1.1" -- PC Magazine, Nov. 1994
That's because you're looking at some MathML code. What one actually types into Mathematica, and sees in Mathematica (or sees in a raw text file version IF the "InputForm" of the code is looked at) is the following. Unfortunately, the code ends suddenly because slashcode somehow doesn't allow more to be shown. BAAAAAD slashcode.
Complaining about the readability of what you posted is like complaining about the raw HTML which goes into this webpage.
[
As part of my undergrad education. Taking less than a second on today's hardware is nothing spectacular; the secret is in the algorithm: You rate the squares according to the number of moves available from that square and, when given a choice, pick the square with the least number of moves. This way, you don't work yourself into a dead-end situation as frequently. Combine this with a little backtracking, and you've got a nice example to show how algorithm selection has a much larger impact on runtime performance than language selection.
Incidentally, 200 MHz was considered a fast CPU when I did it, and I remember it taking 8 billion moves and all night without finding a solution. Until, that is, we implemented the preferential choice part of the algorithm. After that, it was pretty much instantaneous.
The society for a thought-free internet welcomes you.
http://anthonyf.wordpress.com/2006/07/07/solving-the-knights-tour
I think even if you didn't know any lisp you would find this solution to be pretty readable.
Intron: the portion of DNA which expresses nothing useful.
Not to stir up old debates again, but if you like Lisp, you might be better off going for Ruby than for Python. Coming from a Scheme background, I find Ruby to be the more elegant language.
Python is a great language, but my feeling about it is that it's designed to support one way of programming (and not even completely - it's sort of ambivalent between procedural and object-oriented). This is fine, and has the advantage off encouraging consistency among programs from different authors. However, I feel there is a better way: just give programmers building blocks and let the programmers compose them in any way they like. I feel Ruby does this, and the result is a language in which you can elegantly build your program in any way you like. In particular, I feel pure functional style is more natural in Ruby than in Python.
Please correct me if I got my facts wrong.
Not cool.
His code:
# recurse using our neighbours, trying first the ones with the
# least amount of free neighbours, i.e. the "loners"
for ty,tx in sorted(emptyNeighbours, key=lambda c: reduce(
lambda x,y: x+y,
map(lambda j: InRangeAndEmpty(c[0]+j[0], c[1]+j[1]) and 1 or 0,
jumps))):
Fill(ty,tx,counter+1)
Idiomatic python(or at least more-so):
def sort_key(c):
return sum(InRangeAndEmpty(c[0]+j[0], c[1]+j[1]) for j in jumps)
# recurse using our neighbours, trying first the ones with the
# least amount of free neighbours, i.e. the "loners"
for ty,tx in sorted(emptyNeighbours, key=sort_key):
Fill(ty,tx,counter+1)
The nest of lambdas that he wrote the article about isn't the clearest way to write it in python (the reduce( lambda x,y: x+y,...) instead of sum(...) is particularly fun). In my code, wrapping the InRangeAndEmpty in an int() might be preferred, I'm not sure (all that would do is make it clear that the sum is counting the number of squares that are in range and empty).
Nerd rage is the funniest rage.
The ultimate algorithm is called Warnsdorf's heuristic:
http://www.delphiforfun.org/programs/knights_tour.htm
It solves all possible orders (>100x100) in less than a second.
The algorithm cited in the article is really shitty, because it requires recursion.
Hint: I implemented an algorithm to enumerate all magic knight tours (magic, like in magic squares):
http://magictour.free.fr/
Point of fact: Python has the sexiest sprintf() support available. Observe..
>>> print "I ate %d %s in %.3f seconds" % (99,'hotdogs',62.0895)
I ate 99 hotdogs in 62.090 seconds
A non-recursive Python version which uses Warnsdorf's heuristic:
http://github.com/pib/scripts/tree/master/knight.py
It's faster than the one in TFA as well, though it has no backtracking, so it won't find some solutions once you get bigger than 76x76, but at least it doesn't overflow the stack.
It also will tell you whether it found an open, closed, or incomplete path.
TFA has basically stumbled upon Warnsdorff's algorithm. It's a great method, but it doesn't work for really large boards due to blind alleys. There's another method available which uses decomposition to achieve a linear running time(in # of squares), but which is quite a bit more complex to implement. There's a nice tweak to the algorithm which can get much farther than the unmodified original: in the event of a tie (where two candidates have the same number of open neighbours), break the tie by choosing the square farthest from the center. This substantially extends the maximum size of the board (to what, I don't know, because it's worked for everything up to around 450x450, which is past what was described by Arnd Roth).
I won't pretend to remember Lisp inventor John McCarthy's exact words which is odd because there were only about ten but he simply asked if Python could gracefully manipulate Python code as data. "No, John, it can't," said Peter and nothing more, graciously assenting to the professor's critique, and McCarthy said no more though Peter waited a moment to see if he would and in the silence a thousand words were said.
http://smuglispweeny.blogspot.com/2008/02/ooh-ooh-my-turn-why-lisp.html
I like Lisp a lot, and I certainly prefer its syntax to Java or C (or--god help us all--C++), but I still think that it is clearly less readable than Python.
If there's a serious argument against this, it must be Lisp's macro capabilities...
"Not an actor, but he plays one on TV."
Here's code you can put in your Python 2.x code today to future proof it against the change to xrange:
try:
range = xrange
except NameError:
pass
After that, just write range in your code and it will automatically use the equivalent of Python 2's xrange. If you're running Python 2.6, you can use a print function (instead of a print keyword) by adding from __future__ import print_function to your header as well, and you're good to go for a large number of Python 3 switching problems.
But does it work with 3 dimensional chess?
There is an elegant Knight's Tour solver right inside your Python distribution. You can find it at /usr/lib/python2.5/test/test_generators.py. Written by Tim Peters (a.k.a. timbot).
Stop worrying about the risks of nuclear power and start worrying about the risks of not using nuclear power.
Lisp doesn't use CamelCase.
Don't show off your ignorance too much. LISP has used CamelCase for at least thirty years. Admittedly neither Scheme nor Common LISP conventionally use it, but InterLISP certainly does.
I'm old enough to remember when discussions on Slashdot were well informed.