Strategies for Popular Number Puzzles

Arto Inkala Sudoku

There's been some buzz in many news outlets this week [June 2012] about a new puzzle by Arto Inkala, a Finnish mathematician. (for example The Daily Telegraph, The Sun, Metro et al). You can load Arto Inkala's puzzle from this link or pick it from the end of the example list. But is this the hardest puzzle? See below.

We don't have a logical method for solving ALL sudoku puzzles yet so there will be some that defy the pattern based methods used in this solver. Currently, if I produce a large amount of random stock, about 0.01% will still be unsolvable, so it's possible to produce many of these "extremes". People have posted solutions which combine several strategies to get past bottlenecks and there are great ideas I'd love to include, time permitting. The problem is candidate density. If you look at my solver when it comes to 'Run out of known strategies' you will see most cells contain 3 or more candidates. Most of the advanced strategies and all the chaining ones require bi-value (2 in one cell) or bi-location (2 in one unit) to get anywhere. So there is plenty of room for more thought and ideas, which is the attraction of Sudoku — it's very deep.

I've been looking at a new idea for measuring the difficulty of very hard puzzles - ones that can't use the standard scoring because they don't complete. The method is simple like all good ideas. One counts the number of unsolved cells that - if magically filled - render the remaining puzzle trivial. Obviously one counts the insertions separately, not several in one go. Trivial is defined as using Singles, Pairs, Triples, Quads and Intersection Removal, the basic strategies. The ratio of insertions that trivialise a puzzle to those that do not is the score.

But there will be some very hard puzzles where no single insertion makes the puzzle easy. For these level-2 puzzles two cells will normally do the job. Unlike level-1 puzzles where we test 50 to 55 cells the number of combinations of 2 cells is quite high, roughly 1300 to 2100 so it is likely that some or many will trivialize the puzzle. I have yet to find a level-3 puzzle but it will be a truly awesome puzzle if found. Check out the full article for more.

Now, where does Arto Inkala's puzzle fit in in the pantheon of the truly hard? Well, currently third place. David Filmer is the hands down winner with these two puzzles:
  1. Unsolvable #28 just 9 out of 1711 (0.526%)
  2. Unsolvable #49 with 24 out of 1711 (1.4%)
  3. Arto Inkala's puzzle with 79 out of a possible 1770 pairs of cells giving a 4.5% 'trivialization' rate.
  4. Escargot with 80 out of a possble 1596 pairs of cells gives a 5.0% 'trivialization' rate and is fourth
All the remaining contenders I've tested - my stock of unsolvables and extremes - contain between 5% and 30% of pairs.

For a puzzle to have a mere 9 pairs out of 1711 is very interesting and definitively points to Level-3 puzzles. I don't pretend this scoring method is as sophisticated as some more mathematical methods, but as a rough and ready guide, I think it's helpful.

So there you have it. Love to hear your comments and your experience of Arto's monster.

Andrew stuart



... by: Beng

Sunday 5-Nov-2023
Here is the solution for #28 that obtained by my code:

6 2 4 7 8 9 1 3 5
1 7 9 4 5 3 8 2 6
5 8 3 1 2 6 7 4 9
3 4 8 6 1 7 9 5 2
9 6 5 8 4 2 3 7 1
2 1 7 3 9 5 6 8 4
7 9 6 2 3 4 5 1 8
8 5 2 9 7 1 4 6 3
4 3 1 5 6 8 2 9 7
It takes much less time and steps than the Inkala's

... by: Mahadev Kulkarni

Tuesday 28-Feb-2023
I solve newyork times hard and medium level suduko almost every day. Even hardest (devil level) i could solve once at But struggling with this hardest suduko puzzle ever. Is there any step by step video where we can refer n understand the tricks n tactics to solve this suduko.

Also Is there any way we can solve this suduko online like newyork times one. Pls make it available online so that we can try few trial n error steps. 🙂 I googled, not found any online filling one. I have written it to paper n trying.


... by: pwrgreg007

Friday 27-Jan-2023
Hi Andrew,
I've been working on Sudoku puzzles for almost 20 years, and just came across the Arto Inkala puzzle. I am curious if anyone can provide the first step to solve this puzzle. Your solver will not even give a hint, which means that the techniques I use, and ones you've described here on your site, won't work.
Thanks - Greg
Andrew Stuart writes:
We would all like to know that ! :) - [Del]
Add to this Thread

... by: Philip T (Timaru, NZ)

Thursday 21-Feb-2019
It took me a couple of weeks trudging back roads to do this one. Isolating 14 options and tackling only 6 to pull the solution out, is hardly using my mental utensil properly but its out!

... by: mcr

Sunday 11-Dec-2016
I do one sudoku a day, at the "maelstrom" level, and as often as not I need one hint to complete it within 15-20 minutes. In other words, I am an average or below-average solver. But I solved the Inkala puzzle on this website, within 29 minutes. The "world's most difficult" sudoku?

... by: Ellie

Sunday 21-Feb-2016
Solved Arto's 'monster' in about 2 hrs. Let's see if your top two puzzles are any more challenging.

... by: Jan du Plessis (New Zealand)

Sunday 22-Nov-2015
Greetings Andrew,
My awareness of "Arto Inkala" occurred a few hours ago. Brilliant.
Comments/ Questions
- Random Enthusiast (26/9/15) - By now you may already know that clue "5"-block "8" in the original puzzle has been transposed in your two solutions.

With the aid of solution count "0" it took 3 prompts to solve the puzzle.No satisfaction or euphoria experienced doing this.The main aim was to quick core test the solution for a perfect fit. Block "5" and four corner blocks solution output serve as a 45 clue input. The result of this particular 45 clue input is four possible solutions for the row of middle blocks.A perfect core would of course force the completion of the 36 outstanding clues.

"Block fitting" instead of single clue fitting may be too much of a broad side solution attack for your strategy considerations.(?).

NOT "Arto Inkala" directly related but info could help indirectly in this and other situations:

In pattern development exercises solution counts of "13" and less arose but "strategies" managed to find a solution.

A solution count '3" with no strategy solution arose in a result set out below:
possibilities possibilities possibilities
G6 4, 7 G9 4, 7
H3 7, 8 H9 7, 8
J3 7, 8 J6 4, 7 J9 4, 8

On inspection the count should be "2" in my opinion.

The solution counter shows a number for possible solutions - taking strategy elimination steps with no other input whatever sometimes results in " Oops" situations.(?). - conflicting theories. (?).

Can your solution counter limitation be increased from the current "over 500" to say "over 10368"?.It will benefit me,- as a count of one- as an improved Sudoku study aid site.
Said/Asked enough - Jan.

... by: glank

Sunday 27-Sep-2015
I recently wrote a simple solver capable of solving any proper puzzle I've put into it thus far (including Arto Inkala and Unsolvable #28) with pure logic. You can see it in action here:
Let me know if it is of any interest, I'll provide some details about what it does.

... by: Random Enthusiast

Saturday 26-Sep-2015
I used my own solver to do some trial and error and managed to find 2 different solutions for this puzzle. Maybe it's just me.
815 273 694 869 412 357
923 684 157 523 678 194
674 591 283 174 593 286

152 937 846 952 367 841
369 845 721 316 845 729
487 126 935 487 129 635

241 759 368 231 754 968
738 462 519 748 936 512
596 318 472 695 281 473
Andrew Stuart writes:
Missing clue in H4? Should be 5 as per puzzle - [Del]
Add to this Thread

... by: Arthur Allen

Wednesday 12-Aug-2015
I am not a genius.

I consider myself an intelligent individual certainty capable of understanding the most basic to the very complex conditons that occur while attempting to solve the most difficult Sudoku puzzle.

I have toyed with such puzzles for many years.

I solved Arto Inkala's puzzle (21 constants provided) in less than 17 minutes.

Should this be considered an accomplishment in the vast Sudoku problem-solving world?
Andrew Stuart writes:
I'd love to know the steps you took at the crucial junctures.
There is a lot to be said for human intuition and your hunches, if correct and followed through will often break through the bottlenecks. That's certainly the mark of a very good sudoku solver. But by coding the strategies I know into the solver means I can't include such intuition - I can only program purely logical pattern based moves. Most of these derive from such initial intuitions. Can you give me a hint?
- [Del]
Add to this Thread

... by: Traci O

Monday 3-Sep-2012
Thank you for a great Arto Inkala's puzzle. I just finished today, Sept 3. I began on July 2. It took me 153 tries before I got it.

... by: Henry E. Nass (New York City)

Monday 27-Aug-2012
Dear Mr. Stuart,
I am very curious about the new sort of super difficult Sudoku puzzles. Is it so for the latest one, or some of the earlier ones created by this same Finish mathematician, that there is only one order of filled in squares that will solve it ? I suppose its possible that this could be the case but don't know how one would know. I believe it is the case that it is case that there is only one order which will undo the puzzle, but what would determine that ? Of course by analogy, a combination lock has only one order that will unlock it , even if it opens which just 3 or 4 numbers being necessary. Could such a limitation fit in this case ? I would think it would be quite interesting to know, if the people who are able to solve it, follow the same order of solution, at least for the first dozen fill in or so. It seems that would be easy enough to determine if there was a sight where the solving of the sudoku were done on line, and the order of the answer was kept track of, either in real time or by applet. Might one of your readers be able to set this up ? Please let me know I'd like to follow the result of any such study. Thanks. HN
Article created on 30-June-2012. Views: 158209
This page was last modified on 30-June-2012.
All text is copyright and for personal use only but may be reproduced with the permission of the author.
Copyright Andrew Stuart @ Syndicated Puzzles, Privacy, 2007-2024