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The Relative Incidence of Sudoku Strategies

This article has been updated December 2013 and replaces the statistics done in Dec 2009 and March 2012 (you can compare the previous data on that page).

A recent question from a reader prompted me to run off some statistics which I think are interesting and worth exploring.

Comment:There is something I am curious about that I really hope you can answer, although it's quite subjective and I suppose the answer will be a ballpark figure but I was hoping a Sudoku expert such as yourself could take your best educated guess at.

If I know all of your basic, tough, and diabolical strategies, but don't go as far as any of your evil strategies that you list, what percentage of Sudoku puzzles (in your opinion) do you think I could solve-80% of all puzzles that I would try? 85%? 90%? 95%? 99%?

What would you guess if you had to estimate? I know it's hard since there are literally trillions of puzzles, but easy, medium, tough, and many diabolical puzzles I can already solve with these current strategies, excluding your evil ones. Do you think the percentage of puzzles where you HAVE to use one or more evil strategies in order to solve the puzzle is a small percentage, perhaps 1%? 2%? 5%? 10%?

Just curious what your opinion is.

There is a lot to grading and scoring a Sudoku puzzle. I've put some thoughts about this into There is not a one to one correspondence between the published grade (or the grade on my solver) and the list of strategies and many factors contribute to the grade. My strategy list is partially subjective in that I choose to label certain strategies as 'tough' for ease of explanation and to show what I consider the best order in which to attack a puzzle. It is an attempt at a 'minimum path'.

It should also be noted that because I don't use strategy X to solve a puzzle in the solver, it does not follow that strategy X could not be used. There are often many ways to solve the same puzzle.

However it is still an interesting question what proportion of all puzzles require at least one strategy in each grade group. I've run a count on a 120,000 puzzles I created searching for unsolvables (December 2013). These were produced randomly and I did not know the grade until after I created them. The sample is therefore fair. The results are:

  • 54.2% required only trivial strategies, that is only naked and hidden singles.
  • 15.6% required the use of Naked Pairs and Hidden Pairs.
  • 6.6% required the above and diabolical strategies
  • 3.3% required the above and extreme strategies
  • 10 out of 120,000 could not be solved using my list of logical strategies

This confirms my view that the vast majority of puzzles are uninteresting. In order to produce a 100 puzzles of all grades I need to over produce many puzzles since the incidence of higher grade puzzles is low. Note that the 10% of 'moderate' only puzzles does not mean they are rare. Any hard puzzle will require many more incidences of moderate strategies to complete in addition to the hard ones.

It follows that I can produce a list of all the Sudoku strategies and a count of their occurrences in solving the stock. Where different types or rules are available I've also added those as seperate figures.
*Note: The strategy count (larger white numbers) counts puzzles where the strategy is used, not how many times.
The bluish smaller numbers for DO count the number of times the sub-strategy is used.

Naked Singles12000054.23%
Hidden Singles11177150.51%
Naked Pair3107714.04%
Naked Triple15412 6.97%
Hidden Pair6335 2.86%
Hidden Triple1240 0.56%
Naked Quad158 0.07%
Hidden Quad12 0.01%
Tough Strategies
Pointing Pairs22093 9.98%
Line/Box Reduction10969 4.96%
X-Wing5708 2.58%
Simple Colouring12518 5.66%
Rule 24219 23.95%
Rule 4612 3.47%
Rule 512786 72.58%
Y-Wing10363 4.68%
XYZ Wing4143 1.87%
Swordfish835 0.38%
X-Cycle6965 3.15%
XY-Chain12058 5.45%
Diabolical Strategies
3D Medusa3927 1.77%
Rule 1369 6.53%
Rule 2164 2.90%
Rule 3514 9.10%
Rule 4443 7.84%
Rule 51037 18.36%
Rule 62838 50.25%
Rule 7283 5.01%
Jellyfish8 0.00%
Avoidable Rectangle15 0.01%
Unique Rectangle1211 0.55%
Type 1 474 37.21%
Type 2 105 8.24%
Type 2b31 2.43%
Type 3 15 1.18%
Type 3b81 6.36%
Type 4 409 32.10%
Type 4b159 12.48%
Extended Unique Rectangle29 0.01%
Hidden Unique Rectangle2894 1.31%
Type 12126 56.68%
Type 2954 25.43%
Type 2b671 17.89%
WXYZ Wing1448 0.65%
Aligned Pair Exclusion2571 1.16%
Extreme Strategies
Grouped X-Cycle1957 0.88%
Strong Links2659 17.21%
Weak Links11988 77.58%
Off-chain805 5.21%
Empty Rectangle46 0.02%
Finned X-Wing0 0.00%
Finned Swordfish368 0.17%
Franken Swordfish1 0.00%
Alternating Inference Chain4441 2.01%
Strong Links9131 41.18%
Weak Links9532 42.99%
Off-chain3510 15.83%
Sue-de-Coq18 0.01%
Digit Forcing Chain620 0.28%
Nishio Forcing Chain129 0.06%
Cell Forcing Chain610 0.28%
Unit Forcing Chain178 0.08%
Almost Locked Set7 0.00%
Death Blossom1 0.00%
Pattern Overlay15 0.01%
Quad Forcing Chain46 0.02%
Bowman Bingo46 0.02%

So if you were wondering, as I was, how useful certain strategies are, this data is interesting. The only other caveat I'd add is that some strategies are sub-sets of others, or can be expressed in terms of another strategy. For example, Remote Pairs are a special case of XY-Chains which is a sub-set of AICs. It is useful for the solver to split these out but when making and grading I don't do so. So there is some overlap.

The answer to the reader's original question - the incidence of 'evil' strategies, is I'd say, about 5%.

Andrew Stuart

Go back to Arto InkalaContinue to 17 Clue Proof


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Comments Talk

Saturday 3-Sep-2016

... by: Uhm

About Finned X-Wing 0 0.00%

What does this tell, can you run the test and place Finned X-wing before Grouped X-Cycle.

Finned X-wing is a full subset of Grouped X-Cycle
When fin size=1 its a full subset of X-Cycles

Why is there no detail about normal X-Cycles, like "Strong Links, Weak Links, Off-chain " like you did with some other strategies.

About 3d Medusa, there are now only 6 rules instead of 7

Monday 24-Dec-2012

... by: Fred

Very, very interesting ! You have all my compliments and my deepest admiration. Unfortunately I am not of english mother tongue, so for me difficulties always grow harder and harder.
Do you think I can find anything on this subject printed in ITALIAN ? eventually, where ?

Thanks for the attention,

Emmanuele Frediani (from Italy)

P. S. : Just a curiosity : How old are you ?

Friday 29-Jun-2012

... by: slowofmind

Smashblast, could you please say what year is the Scientific American article "The Science behind Sudoku".
Andrew Stuart writes:

2006 I think

Monday 7-May-2012

... by: PeteTy

everything in the grading and strategy seems to be
only relevant to the garden variety sudokus

with jigsaws after pairs and singles
LOL will solve puzzles that most of the chains and advanced stuff cant touch

with killers its a whole new ball game arithmetic gets things that none of the other strategies cant begin to see

things the solver doesnt check for that are very common in killers
example column 1+2+3+4 outie
these seem to jump out at me when i look at a puzzle

sometimes there are triple outies of 6 or 24 that arent apparent to me before the arithmetic but surely help in solving and they give a naked triple that the solvers dont see.

these are simple strategies for killers but beyond diabolical for the garden variety.

Tuesday 1-May-2012

... by: keith

Hmmm. It would be interesting, perhaps, to mix up the order of your solving strategies, and try a series of them, looking for 1) how they change the incidence of use, and 2) whether/how some orders reduce the length of puzzle solving. In the second instance, one can imagine that puzzles "requiring" a diabolical strategy, for example, might benefit from it being applied earlier... or not.

There are a great many orderings you could try, I'm not sure which would be most interesting/powerful, if indeed any succeed over the default...
Andrew Stuart writes:

This I have pondered for a long time and I do try out some reorderings because there is a lot of overlap between certain strategies. Sometimes I can eliminate a strategy altogether by parking it at the end of the list but some are subsets which are easier to identify so I have to put them before the more generic strategy. If I had some time, like some weeks, I'd write a program to test all the combinations and see if there was an optimal arrangement - ie one that reduced the score. The problem with that approach is it might produce a low score for many puzzles but that score depends entirely on following a particular route - which may be esoteric - and therefore the low score is misleading.

The ordering I have settled on is subjective in a certain sense because I've decided on it based on my hunch that strategy A is more complex that strategy B. The early strategies I think are no brainers but diabolical and above are more difficult to order.

I would like to allow the user to order the strategies but I don't know how to do this without greatly complicating the interface - another trade off

Wednesday 8-Feb-2012

... by: Smashblast

There was an eneeellcxt article on the "science behind Sudoku" in July issue of Scientific American, with more strategies, worth reading.

Monday 20-Sep-2010

... by: joseph brophy

thank you so much for your contributions to sudoku. i am preparing to teach a class to seniors, and this is the type of information that sheds light on the game. jtb
Article created on 31-December-2009. Views: 47557
This page was last modified on 10-December-2013.
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