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## Arto Inkala SudokuFrom sudokuwiki.org, the puzzle solver's site |

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.

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:

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:

- Unsolvable #28 just 9 out of 1711 (0.526%)
- Unsolvable #49 with 24 out of 1711 (1.4%)
- Arto Inkala's puzzle with 79 out of a possible 1770 pairs of cells giving a 4.5% 'trivialization' rate.
- Escargot with 80 out of a possble 1596 pairs of cells gives a 5.0% 'trivialization' rate and is fourth

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

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

Andrew stuart