# Pen and Paper (#90)

I enjoyed this problem. It’s fun to play with and gives you a little
room to
get creative with solving techniques.

As many people pointed out, this is pretty much the Knight’s Tour
problem with
some unusual Knight jumps. Most solutions will solve either problem, if
you
change their idea of neighbor squares. The good news about that is that
we can
take advantage of the shortcuts people use to solve that problem.

The most straightforward solution to this problem is I can dream up is:

``````1.  Pick a random starting square
2.  Make a list of all the possible moves from the current square
3.  If there are no moves, undo the last move and try the next choice
from that square
4.  Otherwise, make the first move in the list
5.  Goto step 2 until the grid is full
``````

That’s a boring brute force approach, which is the computer science term
for
“try everything until something works.” When it gets stuck somewhere,
it
backtracks and tries another jump. It may, at times, need to backtrack
several
steps to get back to a place where it could try another move.

This approach has one major plus and one major minus. First, the good
news: it
will find a solution. The bad news: eventually. Because it has to
check every
possibility, it can take a good long while to find a path that works.
The
bigger the board gets, the more places it has to check. The waits get
longer
and longer.

Now, when solving the Knight’s Tour there is a common shortcut that
to a solution much faster. Luckily it works here too. The idea is
that, you
should visit the squares with the least choices first. Those are the
places you
are likely to run out of options, so getting them out of the way while
you still
have plenty of open squares increases your chances of making it around
the
board. This is called the JC Warnsdorff heuristic, because JC
the suggestion all the way back in 1823.

The downside of the JC Warnsdorff heuristic is that is doesn’t always
work.
Depending on your starting position and which squares you visit first,
it can
paint itself into a corner. The problem is more common on certain board
sizes,
but it does happen. The upside is, it’s so darn quick (because it
doesn’t
backtrack), you can do multiple searches and still be quicker than the
brute
force approach. If one attempt fails, just try again. Most solutions
used this
approach.

I found one more corner to cut. When the quiz mentioned circular
solutions, it
made me realize you could cheat a bit, if you had one. If you can go
from the
end of the line back to the beginning (the definition of a circular
solution),
you can start anywhere on that path and follow it for the entire length
to get
all the numbers out. In truth, these are all the same solution (in my
opinion),
but the numbers move around so they look different. It’s also lightning
quick
to shift all the numbers by some offset. Sadly it can be pretty slow to
find a
circular solution in the first place. It’s a trade off.

Another technique, subdividing the grid and piecing together multiple
smaller
solutions, was used be Elliot T… See the later postings in the
for a good discussion of that approach.

OK, let’s get to the code already!

I’m going to so show my solution, just because it takes both shortcuts
and I’m
very familiar with how it works. I do not think my solution came out
the
cleanest though, so definitely go through the others to find some pretty
code.
Here’s the beginning of my solution:

``````#!/usr/bin/env ruby -w

require "enumerator"

class PenAndPaperGame
def self.circular_solutions
@circular ||= if File.exist?("circular_solutions.dump")
``````

}
else
Array.new
end
end

``````  def initialize(size)
@size    = size
@largest = @size * @size

@grid = Array.new(@largest)
end

# ...
``````

I pull in enumerator here, because I’m addicted. Can’t wait until that
library
is in the core.

The class method here is my persistent memory for circular solutions.
the data file, if it exists, and creates an Array of circular solutions
at
various sizes. If we don’t have the file, a new Array is used. The ||=
caching
operator is used for the assignment so we only look the value up once.

The constructor is trivial and almost every solution had one just like
it. We
record the size, figure out what the largest number needs to be, and
build the
grid Array.

The Array was a dumb choice on my part. It somehow made me feel more
manly to
use a one dimensional Array and deal with the two dimensional indexing.
However, looking at David T.s super clear 45 line solution that uses
normal
nested Arrays just made me feel dumb.

Here’s the cheat solver I described earlier:

``````# ...

def solve
if self.class.circular_solutions[@size].nil?
solve_manually
else
@grid  = self.class.circular_solutions[@size]
offset = @grid[rand(@grid.size)]
@grid.map! { |n| (n + offset) % @largest + 1 }
to_s
end
end

# ...
``````

If we haven’t yet found a circular solution for this size, a handoff is
solve_manually(). If we do have one, the else clause is a full
(cheating)
solution. We set the grid to the complete solution, choose a random
value from
the grid (similar to picking a starting square), adjust all the values
in the
grid by the chosen starting point, and print the results.

Now that we’ve looked at my super cheat, let’s get to a real solution:

``````# ...

def solve_manually
x, y  = rand(@size), rand(@size)
count = mark(x, y)

loop do
to = jumps(x, y)
return self.class.new(@size).solve_manually if to.empty?

scores    = rate_jumps(to)
low       = scores.min
next_jump = to.enum_for(:each_with_index).select do |jump|
scores[jump.last] == low
end.sort_by { rand }.first.first

count = mark(*(next_jump + [count]))
x, y  = next_jump

if count > @largest
if circular?
self.class.circular_solutions[@size] = @grid
File.open("circular_solutions.dump", "w") do |file|
Marshal.dump(self.class.circular_solutions, file)
end
else
puts "Found this solution:"
puts to_s
puts "Continuing search for a circular solution..."
return self.class.new(@size).solve_manually
end
end
end
end

# ...
``````

This method looks big, but hopefully it’s fairly high level and thus
easy enough
to read. First, we select a random starting x and y and mark() that
square.
Then we dive into the main solution loop.

In the loop, we pull all possible jumps() from this square and make sure
we have
at least one choice. If we don’t, we got stuck and can’t find a
solution so we
just build a new solver, trigger the search for another attempt, and
return the
results of that.

If we did get some jumps, we need to score them, based on how many moves
we
would have from there. The call to rate_jumps() does this. Then we
pluck out
the low score as our target move. The complicated ball of iterators
right after
that is just a lazy way to select a random move from all the choices
matching
the lowest score, which gets slotted into next_jump.

With a jump selected we mark the new square and move.

Before we loop(), we check to see if that was the final mark. When it
is, we
have a solution, but we want to check if it’s a circular solution we
could reuse
for cheating. If it is circular, we add it to the collection and update
our
storage file. Then we show the solution. When it’s not circular, we go
and show it, but trigger another search to see if we can hunt down a
circular
solution.

Here’s the printing code:

``````# ...

def to_s
width  = @largest.to_s.size
border = " -" + (["-" * width] * @size).join("-") + "- \n"

border +
@grid.enum_for(:each_slice, @size).inject(String.new) do |grid, row|
grid + "| " + row.map { |n| n.to_s.center(width) }.join(" ") + "
``````

|\n"
end +
border
end

``````# ...
``````

Nothing too tricky here. We find a cell size and build a border. Then
we print
a border, each row, and another border. The complicated row iteration
is just
another sign that I used the wrong Array.

Here are the missing helper methods:

``````  # ...

private

def at(x, y)
x + y * @size
end

def mark(current_x, current_y, mark = 1)
@grid[at(current_x, current_y)] = mark
mark + 1
end

def jumps(from_x, from_y, grid = @grid)
[ [-3,  0],
[3,   0],
[0,  -3],
[0,   3],
[2,   2],
[-2,  2],
[2,  -2],
[-2, -2] ].map do |jump|
[from_x + jump.first, from_y + jump.last]
end.select do |jump|
jump.all? { |to| ([email protected]).include? to } and
``````

grid[at(*jump)].nil?
end
end

``````  def rate_jumps(choices)
choices.map { |jump| jumps(*jump).size }
end

def circular?
grid                       = @grid.dup
grid[grid.index(@largest)] = nil

x, y = grid.index(1).divmod(@size).reverse

not jumps(x, y, grid).empty?
end
end

# ...
``````

The at() method is used to turn x and y coordinates into an index for
the one
dimensional grid I am using. mark() will place a number in the
indicated square
and return the next mark that should be placed. (This was another odd
choice.
I have no idea why I didn’t use an instance variable here.)

The jumps() method uses offset to locate all possible moves from the
current
location. It then filters those by removing any out-of-bound indices
and any
squares that aren’t nil. rate_jumps() is just a thin shell over jumps
to count
the moves available at each step.

The circular?() check duplicates the solved grid, knocks the last move
back to
nil, locates the starting square, and checks to see if the starting
square now
has one jump (to the only nil square).

Using those pieces, here’s the application code:

``````# ...

if __FILE__ == \$PROGRAM_NAME
size = ARGV.first && ARGV.first =~ /\A-s(?:ize)?\Z/ ? ARGV.last.to_i
``````

: 5
puts PenAndPaperGame.new(size).solve
end

That just reads the size parameter or sets the default to 5 and kicks
off the
solver process.

A big thank you to all those who were so creative with their solutions
this time
around. You guys taught me how to finish off my own solution.

Tomorrow we will play around with interactively defining Ruby methods…

On Aug 17, 2006, at 8:51 AM, Ruby Q. wrote:

board. This is called the JC Warnsdorff heuristic, because JC
the suggestion all the way back in 1823.

So what I was calling the one-step look-ahead is properly called the
JC Warnsdorff heuristic. I’m glad to learn that.

force approach. If one attempt fails, just try again. Most
solutions used this
approach.

Re: … because it doesn’t backtrack …

The heuristic can be used to improve a backtracking solver. The
interesting question is whether backtracking is ever better than
restarting. I think there are such situations. One might write a
solver that tries backtracking a little first and then does a restart
as Plan B. Of course, one would have to make a precise definition of
what “a little” means

Regards, Morton

On Aug 17, 2006, at 9:23 AM, Morton G. wrote:

force approach. If one attempt fails, just try again. Most
solutions used this
approach.

Re: … because it doesn’t backtrack …

The heuristic can be used to improve a backtracking solver.

That’s a good point. David T.'s solution does both.

James Edward G. II

Just one last minor point. A Google search on Warnsdorff revealed
that it’s HC Warnsdorff, not JC, and that its usually referred to as
the Warnsdorff rule, not Warnsdorff heuristic. The latter is probably
because “rule” is a lot easier to type than “heuristic”

Regards, Morton

On 8/18/06, James Edward G. II [email protected] wrote:

On Aug 17, 2006, at 8:05 PM, Morton G. wrote:

Just one last minor point. A Google search on Warnsdorff revealed
that it’s HC Warnsdorff, not JC

This is a wild shot but the H and the J keys are quite close on the
keyboard. HC gets 299 hits and JC only 113.
Now what intrigues me is that I could not find what
[H299/412,J113/412]
means. So I guess as a native speaker that
HC means Hans Conrad or something similar, but I cannot rule out JC

Interesting. It seems to appear both ways:

I choose to call it a heuristic because I defined those in last
week’s summary. I thought it would be good for the regular readership.

James Edward G. II

Deux choses sont infinies : l’univers et la bÃªtise humaine ; en ce qui
concerne l’univers, je n’en ai pas acquis la certitude absolue.

• Albert Einstein

First post, then think.

But just to tell you it is HC.

My proof goes here
No hits
but
64 hits (can you believe it 64 hits for the Warnsdorff algorithm, that
is
really NICE!!!).

Cheers
Robert

On Aug 17, 2006, at 8:05 PM, Morton G. wrote:

Just one last minor point. A Google search on Warnsdorff revealed
that it’s HC Warnsdorff, not JC

Interesting. It seems to appear both ways:

and that its usually referred to as the Warnsdorff rule, not
Warnsdorff heuristic. The latter is probably because “rule” is a
lot easier to type than “heuristic”

I imagine it’s because it was stated before computer’s existed to
make heuristic a popular term.

I choose to call it a heuristic because I defined those in last
week’s summary. I thought it would be good for the regular readership.

James Edward G. II

On Aug 17, 2006, at 10:23 AM, Morton G. wrote:

One might write a solver that tries backtracking a little first and
then does a restart as Plan B. Of course, one would have to make a
precise definition of what “a little” means

I have implemented such a program, one which applies the Warnsdorff
rule starting from a random position and which allows backtracking to
be turned on or off. I find the results convincing – backtracking
does have value even when the Warnsdorff rule is used, especially
when a cyclic solution is wanted.

17x17 grid – cyclic solution required – average of 5 runs for each
case

no backtracking: 10.2562504 sec
with backtracking: 1.1582514 sec

I appologize if I’m being a bore on this subject, but I hope at least
a few people will be interested in this result.

Regards, Morton

On Aug 18, 2006, at 3:51 AM, Robert D. wrote:

But just to tell you it is HC.

Thanks for looking into this. I’ve updated the summary.

James Edward G. II