In this weekend’s edition of the Sydney Morning Herald, they introduced and over-hyped a type of logic puzzle called Sudoku.

It is a logic puzzle with fairly simple rules. There is a 9×9 grid, broken down into nine 3×3 blocks. The aim is to layout the digits 0 to 9 such that they follow some simple constraints – no row, column or block may have a repeated digit.

You can find far a better description at the Sudoku home page.

Okay, so a new type of puzzle? I’m up for that. Earlier today, I wrote about my approach to tackling such puzzles.

Sudoku turns out to be a similar type of puzzle to the unfortunately-named Logic Problems class of puzzles – you know, the type with clues like “The butcher’s wife plays cribbage with the redhead on Thursdays.” I have seen this type of problem solved by some software written by a colleague in 1992 as he experimented with the Prolog programming language.

As I read the hype about the puzzle, I thought the puzzle would probably fall down to a similar method – which may, or may not, end up being brute-force – I could never be sure with Prolog software.

Then something in the article caught my eye. Wayne Gould, a sudoku expert and owner of the Sudoku home page, chided a beginner:

“So you tried trial and error, then? That’s not the way to go. One golden rule of sudoku is never guess.”

Never guess? That’s an excellent sign from the perspective of solving it automatically. However, a terrible thought occurred to me. Could it be that this entire puzzle, despite all of the hype, is merely using simple elimination, with no higher thought processes? Could the whole thing be that straight-forward?

Let me give an example.

Suppose we have five grid cells in a row A, B, C, D and E which must be assigned unique values 1, 2, 3, 4 or 5.

Suppose we already know some constraints:

A is NOT 3, 4 or 5

B is NOT 3, 4 or 5

C is NOT 5

D is NOT 1 or 5

E is NOT 1

Now, we know that one of the cells must have the value 5, and only E can have the value 5, so by elimination on E must equal 5.

That’s simple.

It takes a little bit more thought to see that A and B both can only be 1 or 2, so therefore no other cell can have the value 1 or 2 – that would mean that there wasn’t room for both A and B.

We can conclude that:

A is NOT 3, 4 or 5

B is NOT 3, 4 or 5

C is NOT1, 2 or 5

D is NOT 1, 2 or 5

E is5

You could draw this same conclusion by backtracking; guessing that D = 2 would eventually lead to the same conclusion. However recognising this pattern generally is a bit harder than simple elimination.

For all I knew, the general Sudoku problem *could* require brute-force to solve (i.e. it could be NP-complete). Alternatively, they *could* require more sophisticated logic than simply ticking off items until a conclusion could be drawn.

However, it could be that the subset of puzzles that actually get printed in the newspaper might not be so hard. The hype just seemed too much; perhaps the printed puzzles were actually trivial.

The weekend SMH ones are supposed to be the hardest ones they publish. If my theory was right, even the hardest published ones would fall easily – not to brute-force guessing. but to methodical elimination.

So away I went.

I wrote 150 lines of Python.

The design consists of three classes, Cells, Neighbourhoods (representing rows, columns and blocks) and the Big Grid.

The Big Grid is just a container which creates the cells and neighbourhoods, and helps them link together, so each cell knows of its three neighbourhoods, and each neighborhood knows of its nine cells.

Each cell is responsible for tracking what values are still legal for that cell. If it every works out by eliminiation what the correct value was for that cell, it notifies its three neighbourhoods of the happy news.

Each neighbourhood is responsible for two tasks – (i) informing the unsolved cells of values that are no longer legal (because they were used by others cells in the neighbourhood) and (ii) watching out for the case that only one cell in the neighbourhood could have a particular value, meaning it could be resolved by elimination.

Notifications go back and forth between the cells informing their neighbourhoods of news, who in turn notify the neighbouring cells, who themselves draw further conclusions and call back to the neighbourhoods.

The system was very recursive – the recursion tails out when a cell is informed of a fact of which it is already aware. Cells also unsubscribe to notifications once they are resolved.

An optimisation would be to push all notifications on a stack and then iterate through them rather than recursing. With this technique, it would be possible to prioritise notifications (“You are a 6! You can stop listening now.” is a more important message than “You are not a 3.”), to consolidate them (“Here is a bunch of messages at once for this cell.”), and to dismiss them faster (“This cell is no longer subscribing – discard the message.”)

However, the speed of the application was not slow, and these optimisations were not required.

So that was my experimental method – what were the results?

An empty grid has 91 cells. A total of 31 clues were provided.

After entering 29 of the clues, the software had still failed to fill in additional cells. After entering 30 of them, it was able to work out only a few of the cells.

After entering the final clue, it worked out a few more – a total of 12 cells of a possible 50.

My hypothesis was wrong: Sudoku is not a trivial puzzle, relying only on elimination.

Only after getting this far, did I let myself search the web to see what other Sudoku solvers are out there.

The first I looked at was Sudoku Solver (I didn’t install it. Install at own risk.)

The techniques Sudoku Solver uses are not surprising. The manual states:

The system will then attempt to solve the puzzle deterministically. For most puzzles this will yield the solution. For some puzzles the deterministic capability produces no further progress, either because the puzzle has more than one solution or because it is very subtle. At this point the system does a structured exploration of solutions, looking for inconsistencies until the solution is found.

A far more interesting site is the similarly named Sudoku Solver by logic.

While this solver, too, will resort to backtracking (They call it “guess-and-check”.) where necessary, the authors seem to have a similar aversion to it. Their web-site explains:

There’s an argument as to whether [guess-and-check] is ‘logic’, so we have made it an option using a tickbox.

Even with backtracking turned off, their software quickly solved the puzzle in the SMH that had thwarted my software. The solution even showed that my half-finished initial practice attempt at solving the same Sudoku manually had a mistake in it.

The authors take the time to explain their methods. They seem to have neglected to mention the most obvious method which they have clearly implemented. (i.e If a cell has eight unique neighbours, it must be the only unused number.) Using their terminology, I only implemented that obvious method, plus Method A. My example above, demonstrating more sophisticated logic would have been cracked by their Method C.

They also admit that they haven’t completely cracked it, and offer you the chance to help – take a stab at one of the solutions they can’t solve automatically, describe how you solved it, and they will try to implement that method.

An exciting idea, until a horrible thought occurs – what if the problem is NP-complete?

Sure enough, the Wikipedia article on Sudoku confirms the bad news: Sudoku is NP-complete. Elegant logic may speed up the processing a little, but when it all boils down to it, you are going to have to try out values one by one until you find one that works.

Wayne Gould, you’re wrong! The golden rule of sudoku is, sometimes, guessing is the only way forward.

Comment by Reader on July 15, 2005

Sudoku made easier at http://www.urbanrim.org.uk/sudoku.htm

Comment by sam @ dailysudoku on July 23, 2005

Ah — I believe you’ve misunderstood his point slightly. Given enough compute, I guess you could automatically generate all possible sudoku puzzles. Only a small subset would have a single solution, and a smaller subset still would allow you to get to that solution entirely logically (no trial and error). Gould’s definition of a valid sudoku (one I subscribe to) requires this logic-only single solution. So, as long as you have one of these “valid” puzzles, guessing is indeed not the way forward.

Sam

Comment by DJ on July 23, 2005

Many of the sites on Essential Links to SuDoku http://www.el.com/links/sudoku.asp are quite expert and some recommend guessing as a strategy. Perhaps it isn’t really necessary.

Comment by gr8nsitian on August 9, 2005

hey ppl…………

im trying to write code for sudoku……..

its amazing tht i wrote complete code in 1day only bt in the last it found to b wrong.:(

can nebody help me out about logic of genrating sudoku……

gr8nsitian

Comment by Olivier Verdin on September 21, 2005

Check out http://www.sudokuprime.com and play sudoku with your friends in a multi-user session. Have fun!

Comment by John Hogan on October 4, 2005

I have written a program to solve Sudoku w/o guessing. So far I haven’t found a puzzle that it can’t solve. Please show me a puzzle that has a unique solution and requires guessing so that I can try it out. Thanks

Comment by Arnaud on December 6, 2005

I’m french and i think it’s a very good game for logic. So i have made a web site about sudoku : http://www.e-sudoku.fr.

Comment by Aristotle Pagaltzis on February 20, 2006

It’s so good that it has to be comment-spammed, huh?

[Editor’s note: The comment Aristotle was referring too has since been deleted. It was spruiking yet another Sudoku page. It was a hard call to label it spam. The poster passed a CAPCHA test. There has been a precedent too; there have been a few comments above that have simply recommended Sudoku pages that I have let through, but I think Aristotle is right. That’s enough links that don’t actually contribute to the discussion.]

Comment by Siddharth on May 14, 2006

Can any of you guys email me the code? I’m trying to write the code myself. But I really haven’t got the time yet. I intend to write it soeday…

Comment by Aristotle Pagaltzis on May 26, 2006

Peter Norvig takes on the problem.

Comment by Ganesh Jaju on July 17, 2006

Hey ,

I want to make a point that we can solve Sudoku

deterministicallyEarlier i solved some problems by resorting to guesses and backtracking as n when required

But soon I found that we can solve it w/o any guesses

Eliminationis the best method.Each time you have fixed the position of a digit ,there is some other whose position has become deterministically predictable and so you can continue in this manner untill you are done

Comment by Piggy on August 16, 2006

Re Ganesh and others, I think there is a convention in the newspaper puzzles–this is basically rephrasing Sam’s sets.

-Easy: can be done with simple elimination.

– Medium: can be done with multi-step elimination.

– Hard: there is a unique solution but it cannot be found with elimination alone, therefore at least one guess-and-backtrack is necessary.

Comment by wossname on July 24, 2007

.8…..2.

..1…6..

2…5…3

..65.12..

7..6.4..9

..47.93..

6…1…5

..7…9..

.4…..3.

My Deterministic solver fails at solving the above. Can anyone solve prove me wrong?

Comment by trobin on August 7, 2007

This message is for “wossname”

.8…..2.

..1…6..

2…5…3

..65.12..

7..6.4..9

..47.93..

6…1…5

..7…9..

.4…..3.

I used my own “ELIMINATION” method to solve this problem in aproximately 25 minutes without resorting to guessing on a pair. Would love to find a publisher that would be interested in this method/grid that I have developed. Any suggestions??

Comment by Julie Lawrence on April 20, 2008

Hey Julian, isn’t Sukoku just a variant on Latin Squares and Combinatorial Designs? I did my never-completed PhD on algorithms for completing combinatorial designs, but can’t remember enough about it (and haven’t bothered trying) to be sure that I’m right.

Latin Squares, as I’m sure you know, are n x n grids which can only have the elements 1..n appearing once in each row. Combinatorial designs are ways of choosing a set of sets of 1..n elements, such that each element occurs a set number of times. eg a 2(9,3,1) design is a bunch of 3-sets, such that each element occurs in exactly two of the 3-sets

A combinatorial design has one or more “defining set”, being a partial design which can only be completed in one way (and a “minimal defining set” being the smallest such set) i.e. the entries you start off with in a sudoku puzzle are simply a defining set.

Anyone know anything more about whether these things are interchangable?

julie

Comment by Jack Eisenhover on March 24, 2009

Dear Editor

while looking for simple solutions and explanantions for Sudoku solvers, I came up with the following site

http://alumni.cs.ucsb.edu/~vijay/Sudoku.html

It seems to explain solutions easily and claims to solve evil problems.

Personally, I find Sudokus a gross waste of time.

Jack-E

Comment by super sudoku solver on November 19, 2009

a waste of time.. 🙂 that’s blasphemy! I can’t gte enough these days.

Comment by Website optimisation on January 4, 2010

Soduku is fun but time consuming

Comment by treatment for sciatica on November 29, 2011

In reply to the comment a waste of time.. Sudoku is very old now and still even after all these years so very popular, kind of speaks for it self I think