-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- The three rules of Ruby Q. 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 Q. 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 T. follow the discussion. Please reply to the original quiz message, if you can. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- ## Sudoku Generator (#182) _Quiz idea provided by Lloyd L._ 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

on 2008-11-07 17:05

on 2008-11-10 19:28

> > ## Sudoku Generator (#182) > > > _Quiz idea provided by Lloyd L._ > > 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. If the requirement to create puzzles of specified difficulty (or determine a puzzle's difficulty) is too much, feel free to simply generate a puzzle.

on 2008-11-11 07:37

On Mon, Nov 10, 2008 at 12:25 PM, Matthew M. <removed_email_address@domain.invalid> wrote: >> ## Sudoku Generator (#182) >> >> _Quiz idea provided by Lloyd L._ >> >> 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. > > If the requirement to create puzzles of specified difficulty (or determine a > puzzle's difficulty) is too much, feel free to simply generate a puzzle. I haven't take the time to look at difficulty level, but have a very naive generator; the basic idea is to generate a puzzle using a sudoku solver and then punch some random holes in it: #!/usr/bin/env ruby # so far, only sudoku-x has completed in the time I'm willing to wait :-) #require 'small_sudoku_solver' #require 'yet_another_sudoku_solver' require 'sudoku-x' require 'enumerator' puzzle = [0] * 81 a = (1..9).sort_by{rand} b = (1..9).sort_by{rand} c = (1..9).sort_by{rand} puzzle[0..2] = a[0..2] puzzle[9..11] = a[3..5] puzzle[18..20] = a[6..8] puzzle[30..32] = b[0..2] puzzle[39..41] = b[3..5] puzzle[48..50] = b[6..8] puzzle[60..62] = b[0..2] puzzle[69..71] = b[3..5] puzzle[78..80] = b[6..8] puts "Seed Puzzle" puzzle.each_slice(9){|s| p s} puts puzzle = solve(puzzle) puts "Solved Puzzle" puzzle.each_slice(9){|s| p s} puts 72.times{puzzle[rand(81)] = 0} puts "Puzzle" puzzle.each_slice(9){|s| p s} puts Full output: Seed Puzzle [6, 8, 5, 0, 0, 0, 0, 0, 0] [3, 1, 9, 0, 0, 0, 0, 0, 0] [7, 2, 4, 0, 0, 0, 0, 0, 0] [0, 0, 0, 2, 1, 8, 0, 0, 0] [0, 0, 0, 4, 5, 6, 0, 0, 0] [0, 0, 0, 9, 7, 3, 0, 0, 0] [0, 0, 0, 0, 0, 0, 2, 1, 8] [0, 0, 0, 0, 0, 0, 4, 5, 6] [0, 0, 0, 0, 0, 0, 9, 7, 3] Solved Puzzle [6, 8, 5, 7, 3, 9, 1, 2, 4] [3, 1, 9, 6, 2, 4, 5, 8, 7] [7, 2, 4, 1, 8, 5, 3, 6, 9] [9, 4, 6, 2, 1, 8, 7, 3, 5] [1, 3, 7, 4, 5, 6, 8, 9, 2] [2, 5, 8, 9, 7, 3, 6, 4, 1] [4, 9, 3, 5, 6, 7, 2, 1, 8] [8, 7, 1, 3, 9, 2, 4, 5, 6] [5, 6, 2, 8, 4, 1, 9, 7, 3] Puzzle [0, 8, 5, 0, 3, 9, 0, 2, 0] [0, 0, 0, 6, 0, 0, 0, 0, 0] [0, 0, 4, 1, 0, 5, 3, 0, 0] [0, 0, 0, 2, 0, 8, 0, 0, 5] [0, 3, 0, 0, 0, 6, 0, 9, 2] [2, 5, 8, 0, 0, 0, 0, 0, 0] [4, 9, 0, 5, 0, 0, 2, 0, 0] [0, 7, 0, 3, 9, 0, 4, 0, 6] [0, 0, 2, 8, 4, 0, 9, 7, 0] Some more puzzles: [3, 0, 0, 9, 0, 0, 0, 8, 0] [0, 7, 4, 0, 3, 0, 0, 0, 0] [0, 8, 0, 0, 6, 5, 1, 0, 0] [0, 4, 0, 3, 1, 0, 0, 0, 0] [0, 0, 0, 4, 9, 7, 0, 0, 0] [7, 0, 9, 2, 0, 0, 0, 0, 3] [4, 5, 7, 6, 2, 9, 0, 0, 8] [6, 0, 0, 0, 8, 3, 0, 0, 0] [0, 0, 0, 1, 0, 0, 0, 0, 6] [6, 0, 0, 0, 3, 0, 0, 0, 8] [0, 8, 4, 0, 0, 7, 2, 0, 9] [0, 0, 2, 5, 0, 0, 0, 0, 7] [0, 0, 6, 0, 0, 0, 5, 0, 0] [0, 0, 0, 0, 4, 1, 0, 8, 0] [0, 0, 1, 0, 7, 0, 0, 0, 6] [0, 0, 0, 0, 1, 6, 0, 0, 3] [0, 0, 0, 0, 5, 0, 0, 4, 1] [0, 6, 3, 4, 0, 2, 8, 0, 0] [4, 8, 1, 0, 0, 0, 9, 0, 0] [0, 0, 6, 0, 0, 0, 0, 0, 4] [0, 0, 5, 1, 0, 0, 0, 0, 0] [0, 0, 0, 8, 5, 0, 0, 0, 0] [8, 1, 9, 0, 3, 6, 5, 0, 0] [5, 6, 0, 0, 0, 1, 0, 0, 8] [1, 0, 0, 0, 0, 0, 8, 0, 0] [7, 0, 2, 0, 0, 0, 4, 3, 0] [0, 4, 0, 3, 0, 0, 2, 9, 0]

on 2008-11-11 07:41

On Tue, Nov 11, 2008 at 12:36 AM, <removed_email_address@domain.invalid> wrote: > puzzle[9..11] = a[3..5] > puzzle[18..20] = a[6..8] > > puzzle[30..32] = b[0..2] > puzzle[39..41] = b[3..5] > puzzle[48..50] = b[6..8] > > puzzle[60..62] = b[0..2] > puzzle[69..71] = b[3..5] > puzzle[78..80] = b[6..8] :-( obviously I meant to use a,b,c not a,b,b when seeding the board; that might make the puzzles slightly more interesting :-)

on 2008-11-11 08:16

On Tue, Nov 11, 2008 at 12:34 AM, <removed_email_address@domain.invalid> wrote: > On Mon, Nov 10, 2008 at 12:25 PM, Matthew M. <removed_email_address@domain.invalid> wrote: >>> ## Sudoku Generator (#182) >>> >>> _Quiz idea provided by Lloyd L._ > > I haven't take the time to look at difficulty level, but have a very > naive generator; the basic idea is to generate a puzzle using a sudoku > solver and then punch some random holes in it: > > 72.times{puzzle[rand(81)] = 0} Hmm, changed that while I was messing around with it, originally: 64.times{puzzle[rand(81)] = 0} Since, it appears that any (unique) Sudoku puzzle must have at least 17 given numbers: http://people.csse.uwa.edu.au/gordon/sudokumin.php (Though, I take no effort to ensure that the at-least 17 numbers left will lead to a unique solution.)

on 2008-11-11 09:35

On 2008-11-11, removed_email_address@domain.invalid <removed_email_address@domain.invalid> wrote: > I haven't take the time to look at difficulty level, but have a very > naive generator; the basic idea is to generate a puzzle using a sudoku > solver and then punch some random holes in it: [...] > 72.times{puzzle[rand(81)] = 0} In the very unlikely event that all 72 holes go to unique locations, you will end up with an improper puzzle (more than one solution), because it will have only 9 givens. The minimum number of givens for a standard 9x9 Sudoku is believed to be 17. When this is possible, the givens can't be in any randomly chosen 17 places, either. 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. As a rough estimate of difficulty level, you could use the number of givens. I.e. if you manage to produce a puzzle with only 17 givens this way, that could be assigned the highest difficulty level. (Though difficulty doesn't exactly correlate with the number of givens).

on 2008-11-11 16:53

On Tue, Nov 11, 2008 at 2:32 AM, Kaz K. <removed_email_address@domain.invalid> wrote: > > 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. Yes, that was my next step, but I never got around to it this past weekend. The third solver 'sudoku-x' apparently can report multiple solutions (the others are too brute-force to populate the blank board in any reasonable time on any hardware I have). Keeping with the random theme, the next planned iteration was like: 64 times pick a random spot skip it if it is already a hole poke a hole and pass the puzzle to the solver if the solver finds more than one solution put the number back and try another the solver finds one solution leave the number out and try another And the next iteration would replace 64 with a number calculated from difficulty level. (I wasn't going to post at all, but I was afraid from the second post from Matthew M. that he was afraid that no one was interested in the quiz.)

on 2008-11-12 06:15

On 2008-11-11, removed_email_address@domain.invalid <removed_email_address@domain.invalid> 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?

on 2008-11-12 06:34

```
On Nov 11, 2008, at 10:12 PM, Kaz K. wrote:
>
That alone does not a Sudoku make. Unless you've got some other
criteria in mind that you don't mention, that's just a 9x9 grid of
random numbers.
```

on 2008-11-12 15:35

Hi Matthew, On Tue, Nov 11, 2008 at 10:30 PM, Matthew M. <removed_email_address@domain.invalid> wrote: >> exchange method) and then take successive elements from this >> sequence as the locations on the Sudoku board? > > That alone does not a Sudoku make. Unless you've got some other > criteria in mind that you don't mention, that's just a 9x9 grid > of random numbers. I think Kaz meant that you can do without the ugly "pick a random spot... darn! I had already looked at this one!" IOW, shuffle a list of all the numbers between 0 and 80, and pick a random amount of them (1 .. 64) that are removed in order to generate the sudoku puzzle. It's an interesting question whether or not this generates better puzzles. Marcelo

on 2008-11-12 16:58

On Tue, Nov 11, 2008 at 11:12 PM, Kaz K. <removed_email_address@domain.invalid> wrote: > On 2008-11-11, removed_email_address@domain.invalid <removed_email_address@domain.invalid> 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)...

on 2008-11-13 18:14

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 K._ 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

on 2008-11-14 07:40

On Thu, 13 Nov 2008 11:10:54 -0500, Matthew M. wrote: > 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.

on 2008-11-14 15:45

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..8).each{|i| @row_index[i] = Array.new s = i * 9 (s..s+8).each{|j| @row_index[i] << j } } @col_index = Array.new (0..8).each{|i| @col_index[i] = Array.new (0..8).each{|j| @col_index[i] << (j * 9) + i } } @block_index = Array.new block_pattern = [0,1,2,9,10,11,18,19,20] (0..8).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..8).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..8).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 == 8) && (!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..8).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 M. wrote: > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > The three rules of Ruby Q. 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 Q. 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 T. follow the discussion. Please reply to > the original quiz message, if you can. > > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > ## Sudoku Generator (#182) > > > _Quiz idea provided by Lloyd L._ > > 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