Slashdot Mirror
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
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.
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
[
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.
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.
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
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.