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[QUIZ] Sudoku Generator (#182)

 
 
brabuhr@gmail.com
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      11-12-2008
On Tue, Nov 11, 2008 at 11:12 PM, Kaz Kylheku <(E-Mail Removed)> wrote:
> On 2008-11-11, http://www.velocityreviews.com/forums/(E-Mail Removed) <(E-Mail Removed)> wrote:
>> 64 times
>> pick a random spot
>> skip it if it is already a hole

>
> Is it really so difficult to make a sequence of integers from 0 to 80, scramble
> its order (e.g. using the Knuth random exchange method) and then take
> successive elements from this sequence as the locations on the Sudoku
> board?


No, it's not difficult at all. (But maybe less fun Doesn't Ruby
1.9.1 have something like:

(0..80).sample(n)...

 
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Matthew Moss
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      11-13-2008

In this week's quiz, I ended up dropping the requirement to select or
determine puzzle difficulty. That, I suspect, is a much harder problem
than generating a quiz and also somewhat subjective. I even wondered
if anyone would attempt generating _any_ Sudoku puzzles, but _brabuhr_
presented a solution that is almost trivial. Granted, it does require
the use of a Sudoku solver (such as sudoku-x used, or perhaps one from
[quiz #43][1]), but I have no arguments against good reuse of code!

brabuhr begins by generating what is called the _seed puzzle_. This is
a partially-filled puzzle that should be solveable. The code for this
is:

puzzle = [0] * 81

a = (1..9).sort_by{rand}
b = (1..9).sort_by{rand}
c = (1..9).sort_by{rand}

# Completely fill in the upper-left 3x3 section.
puzzle[0..2] = a[0..2]
puzzle[9..11] = a[3..5]
puzzle[18..20] = a[6..8]

# Completely fill in the center 3x3 section.
puzzle[30..32] = b[0..2]
puzzle[39..41] = b[3..5]
puzzle[48..50] = b[6..8]

# Completely fill in the lower-right 3x3 section.
puzzle[60..62] = c[0..2]
puzzle[69..71] = c[3..5]
puzzle[78..80] = c[6..8]

I added in a few comments to show what parts of the 9x9 puzzle are
being modified. As the upper-left, central, and lower-right 3x3
sections are completely independent of one another, they can be filled
at random without any expection of contradiction (assuming the rest of
the puzzle is still empty, ensured here by the initial fill of zero).

Visually, the seed puzzle will look something like this (zeros have
been replaced with blanks to improve clarity):

+-------+-------+-------+
| 6 8 5 | | |
| 3 1 9 | | |
| 7 2 4 | | |
+-------+-------+-------+
| | 2 1 8 | |
| | 4 5 6 | |
| | 9 7 3 | |
+-------+-------+-------+
| | | 2 1 8 |
| | | 4 5 6 |
| | | 9 7 3 |
+-------+-------+-------+

The next step is to generate the rest of the puzzle. But since this is
exactly what a solver does, brabuhr uses a solver to generate the
puzzle.

puzzle = solve(puzzle)

I'm not sure whether or not the seed has multiple solutions, but it
doesn't really matter. This is just the first part of creating a
puzzle for humans to solve, so as long as the solving library provides
_some_ solution, we'll have a usable puzzle.

The final step is to take the "solved" puzzle and poke holes in it,
enough so we have a real puzzle for humans to solve. Again, this is
quite simple:

64.times{puzzle[rand(81)] = 0}

This line will punch at most 64 holes into the puzzle. 64 is chosen as
the upper limit, since there seems to be some evidence that the
[Minimum Sudokus][2] -- puzzles uniquely solveable with the least
number of clues -- seems to require 17 clues (and 81 - 17 = 64). It is
quite likely, however, that there will be some overlap in the hole
choices, and so there will likely be more than 17 clues: fewer holes
means more clues, which means (generally) an easier puzzle.

So it is certainly possible that this generator will create puzzles
with more than one solution. _Kaz Kylheku_ provided a suggestion to
deal with that:

An obvious way to improve your generator would be to call the
solver function
after punching a hole. (The solver function hopefully tells you
that the
puzzle has two or more solutions, right?) If after punching a
hole, the puzzle
has more than one solution, then backtrack; restore the hole, and
punch out a
different number.




[1]: http://rubyquiz.com/quiz43.html
[2]: http://people.csse.uwa.edu.au/gordon/sudokumin.php



 
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Ken Bloom
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      11-14-2008
On Thu, 13 Nov 2008 11:10:54 -0500, Matthew Moss wrote:
> The final step is to take the "solved" puzzle and poke holes in it,
> enough so we have a real puzzle for humans to solve. Again, this is
> quite simple:
>
> 64.times{puzzle[rand(81)] = 0}
>
> This line will punch at most 64 holes into the puzzle. 64 is chosen as
> the upper limit, since there seems to be some evidence that the [Minimum
> Sudokus][2] -- puzzles uniquely solveable with the least number of clues
> -- seems to require 17 clues (and 81 - 17 = 64). It is quite likely,
> however, that there will be some overlap in the hole choices, and so
> there will likely be more than 17 clues: fewer holes means more clues,
> which means (generally) an easier puzzle.


This will punch out, on average, 44 holes.




--
Chanoch (Ken) Bloom. PhD candidate. Linguistic Cognition Laboratory.
Department of Computer Science. Illinois Institute of Technology.
http://www.iit.edu/~kbloom1/
 
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Ken Nishimura
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      11-14-2008
Hi all,

Although I'm quite new to both Ruby and programming, sudoku generator
was the problem I picked to practice the basics over the last couple
of weeks. I just came across this thread by chance, so I thought I
might as well put my code here. I know nothing about algorithm or CS
stuff, so I just used a fairly naive approach.

(1) Fill a matrix using 1-9 in each row, column and block.

(2) Pick one cell, and see if punching the hole there will produce
another solution.

(3) If there's a uniq solution, then punch out the cell. And, repeat
this until you check all the cells.

I found that (1) wasn't as easy as I thought. You need some sort of
good way to do this, but again, this was just my practice of Ruby
programming, so I just brute forced: trial and error.

#!/usr/bin/ruby

=begin
= sudoku
* Data structure
matrix = [1,2,3,4,,,,,,,81]
row_index = [[0,1,2,...8], [9,10,11...17],
col_index = [[0,9,18,...72], [1,10,19...73],
block_index = [[0,1,2,9,10,11,],[3,4,5,12,],,]
* Block numbering
|0|1|2|
|3|4|5|
|6|7|8|
=end

class Matrix
attr_accessor :row_index, :col_index, :block_index, :matrix

def initialize
@matrix = Array.new(81,0)

@row_index = Array.new
(0...each{|i|
@row_index[i] = Array.new
s = i * 9
(s..s+.each{|j|
@row_index[i] << j
}
}

@col_index = Array.new
(0...each{|i|
@col_index[i] = Array.new
(0...each{|j|
@col_index[i] << (j * 9) + i
}
}

@block_index = Array.new
block_pattern = [0,1,2,9,10,11,18,19,20]
(0...each{|i|
@block_index[i] = block_pattern.map{|j|
(j + (i / 3) * 27) + ((i % 3) * 3)}
}
end

def row(x)
@row_index[x].collect{|x| @matrix[x]}
end

def col(x)
@col_index[x].collect{|x| @matrix[x]}
end

def block(x)
@block_index[x].collect{|x| @matrix[x]}
end

def which_block(x,y)
((y / 3) * 3) + (x / 3)
end

def index(x,y)
x + (y * 9)
end

def fill_matrix
srand
100.times{|i|
break if self.try_fill_matrix
}
# average 7.53 times
end

def try_fill_matrix
count = 0
abandon_flag = false

@matrix.fill(0)

(0...each{|y|
repeat_flag = true
break if(abandon_flag == true)
until(repeat_flag == false)
count += 1
if (count > 20)
abandon_flag = true
@matrix.fill(0)
break
end
seeds = (1..9).to_a
(0...each{|x|
appear = col(x) | block(which_block(x,y))
n = (seeds - appear).pick_one
@matrix[index(x,y)] = n
seeds.delete(n)
if((x == && (!row(y).include?(nil)))
repeat_flag = false
end
}
end
}
!abandon_flag
end

def make_new_puzzle
self.fill_matrix
self.reduce
end

def reduce
srand
candidate = (0..80).to_a
candidate.delete_if{|i| @matrix[i] == 0}

while(candidate.size > 0)
c = candidate.pick_one
if(uniq_solution?(c))
@matrix[c] = 0
end
candidate.delete(c)
end
end

def reduce_by_quadruple
srand
candidate = (0..80).to_a
candidate.find{|i| ((i % 9) <= 4) && ((i / 9) <= 4)}

while(candidate.size > 0)
c1 = candidate.pick_one
c2 = 80 - c1
if(uniq_solution?(c1) && (uniq_solution?(c2)))
@matrix[c1] = @matrix[c2] = 0
end
candidate.delete(c1)
candidate.delete(c2)
end
end

def uniq_solution?(n)
i = @matrix[n]
x = n % 9
y = n / 9

(1..9).to_a.delete_if{|n| n == i}.each{|j|
if(!col(x).include?(j) &&
!row(y).include?(j) &&
!block(which_block(x,y)).include?(j))
return false
end
}
end

def to_s
print "-"*19,"\n"
(0...each{|y|
i = 0
row(y).each{|n|
if((i % 3) == 0)
separator = "|"
else
separator = " "
end
n = " " if n == 0
print separator, n
i += 1
}
print "|\n"
if(((y + 1) % 3) == 0)
print "-"*19,"\n"
end
}
end

def to_line
self.matrix.join
end

end

class Array
def pick_one
r = rand(self.size)
self[r]
end
end

m = Matrix.new
m.make_new_puzzle
puts m

This script seemed to generate decent sudoku puzzles like the one
below, and I went further.

-------------------
| 1 | 8 | 5|
|8 7 5|3 9| |
| 3|5 7 |9 4|
-------------------
|4 5 | 2| 3 |
|6 8 |1 5| |
|3 |8 6 |2 5 1|
-------------------
| 2 8|6 1 3|5 |
| 9|4 |6 2 8|
|5 | |7 |
-------------------

Puzzles generated with this thing are not as fun, difficult to solve
at all. All the puzzles were too easy with many hints left. The
numbers of hints are between 35-50, averaging 42.7. Thinking that
maybe symmetry is a key to a good sudoku, I added this method:

def reduce_by_quadruple
srand
candidate = (0..80).to_a
candidate.find{|i| ((i % 9) <= 4) && ((i / 9) <= 4)}

while(candidate.size > 0)
c1 = candidate.pick_one
c2 = 80 - c1
if(uniq_solution?(c1) && (uniq_solution?(c2)))
@matrix[c1] = @matrix[c2] = 0
end
candidate.delete(c1)
candidate.delete(c2)
end
end

This method reduces a set of 4 cells that are in symmetric positions at
a time. The results were somewhat interesting. The numbers remaining
for each approach were:

(a) Reduce one by one : 42.7
(b) Reduce 4 cells at a time : 47.8
(c) Reduce 4 cells at a time, when that ends, reduce one by one: 42.5

You can see apparent symmetry in the puzzles generated with (b) and (c),
yet even (c) doesn't yield any better puzzles at all.

Any given matrix filled with arbitrary numbers ends up either a
symmetric or a random puzzle with almost same amount of hints?

By the way, I was kind of sure that the puzzles generated with this
script are okay because there's nothing complicated involved, however,
I used somebody else's solver to check if any of the puzzles has a
uniq solution. Alas, 3-5 out of 100 puzzles, there were more than 1
solution...

I don't know what I did wrong. I just lost interest and felt content
with the fact that I learnt many things with Ruby and had fun doing
this. And then, I found this thread, couldn't resist.

--
Ken Nishimura, Tokyo


Matthew Moss wrote:
> -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
>
> The three rules of Ruby Quiz 2:
>
> 1. Please do not post any solutions or spoiler discussion for this
> quiz until 48 hours have passed from the time on this message.
>
> 2. Support Ruby Quiz 2 by submitting ideas as often as you can!
> Visit <http://splatbang.com/rubyquiz/>.
>
> 3. Enjoy!
>
> Suggestion: A [QUIZ] in the subject of emails about the problem
> helps everyone on Ruby Talk follow the discussion. Please reply to
> the original quiz message, if you can.
>
> -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
>
> ## Sudoku Generator (#182)
>
>
> _Quiz idea provided by Lloyd Linklater_
>
> A bit over three years ago, we had a quiz to [solve sudoku puzzles]
> [1]. Now it's time to write a script that generates sudoku puzzles.
>
> The output of your script should be the puzzle to solve. (Since we
> already have solver scripts from quiz #43, there is no need to output
> the solution.) In addition to generating the puzzle, you should adhere
> either one or the other of these two methods:
>
> 1. Reduce a generated puzzle to the fewest clues that will still
> suffice for finding a solution. To your output, include an estimated
> difficulty level.
>
> 2. Accept a command line parameter: the estimated difficulty level.
> Generate the puzzle such that it roughly matches that difficulty level.
>
> The difficulty level should be a number from 1 (easiest) to 10
> (hardest). Difficulty level, obviously, is somewhat subjective.
> However, there are [various sudoku techniques][2] that may be able to
> help you decide whether a puzzle is more difficult or not. Some
> suggested metrics include: number of clues, number of "gimmes", number
> of possible solutions, cascading singletons, etc.
>
>
> [1]: http://rubyquiz.com/quiz43.html
> [2]: http://www.sadmansoftware.com/sudoku/techniques.htm


--
Posted via http://www.ruby-forum.com/.

 
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