This article has been updated March 2012 and replaces the statistics done in Dec 2009

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
http://www.scanraid.com/Sudoku_Creation_and_Grading.pdf. 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 200,000 puzzles I created searching for unsolvables (March 2012). These were produced randomly and I did not know the grade until after I created them. The sample is therefore fair. The results are:

• 72.5% required only trivial strategies, that is only naked and hidden singles.
  
• 10.0% required the use of Naked Pairs and Hidden Pairs.
  
• 9.6% required the above and diabolical strategies
  
• 3.6% required the above and extreme strategies
  
• 0.00048% (15 out of 311,000) could not be solved using my list of logical strategies, although 49 did solve using the 'trail and error' Bowman's Bingo.

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 incidence in solving the stock.

STRATEGY	Count	%
Human Strategy	200000	100.0%
Naked Singles	180387	90.2%
Hidden Singles	99185	49.6%
Naked Pairs	45175	22.6%
Naked Triples	21359	10.7%
Hidden Pairs	7942	4.0%
Hidden Triples	1737	0.9%
Naked Quads	249	0.1%
Hidden Quads	0	0.0%
Tough Strategies
Intersection Removal	34692	17.3%
X-Wing	9576	4.8%
Simple Colouring	20030	10.0%
Y-Wings	16043	8.0%
Sword-Fish	1313	0.7%
X-Cycles	11050	5.5%
XY-Chain	16724	8.4%
Diabolical Strategies
3D Medusa	5950	3.0%
Jelly-Fish	24	0.0%
Avoidable Rectangle	22	0.0%
Unique Rectangles	1748	0.9%
Hidden Unique Rectangles	4185	2.1%
XYZ Wing	1539	0.8%
Aligned Pair Exclusion	2313	1.2%
Evil Strategies
Grouped X-Cycles	1883	0.9%
Empty Rectangles	144	0.1%
Finned X-Wing	1150	0.6%
Finned Sword-Fish	826	0.4%
Franken Sword-Fish	6	0.0030%
Altern. Inference Chains	6777	3.4%
Strong Links	6301	3.2%
Weak Links	14514	7.3%
Off-chain	1951	1.0%
Digit Forcing Chains	961	0.5%
Cell Forcing Chains	912	0.5%
Unit Forcing Chains	278	0.1390%
Sue-de-Coq	0	0.0%
Almost Locked Sets	12	0.0060%
Death Blossom	4	0.0020%
Pattern Overlay Method	28	0.0140%
Quad Forcing Chains	52	0.0260%
Bowman Bingo	49	0.0245%

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

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