The Relative Incidence of Sudoku Strategies

This article has been updated May 2024 and replaces the statistics done in Dec 2009 and March 2012 and 2013 (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
https://www.sudokuwiki.org/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 141,672 puzzles I created in May 2024. These were produced randomly and I did not know the grade until after I created them. The sample is therefore fair. The results are:

• 48.4% required only trivial strategies, that is only naked and hidden singles.
  
• 17.6% required the use of Naked Pairs and Hidden Pairs.
  
• 6.5% required the above and diabolical strategies
  
• 4.0% required the above and extreme strategies
  
• In addition to 141,672 solvable puzzles made, 4 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.
*

Naked Singles				141672	2585866	74.48%
Hidden Singles				136422	495774	14.28%
Naked Pair				44959	71858	2.07%
Naked Triple				23542	29360	0.85%
Hidden Pair				9194	10179	0.29%
Hidden Triple				1882	1932	0.06%
Naked Quad				255	258	0.01%
Hidden Quad				13	13	0.00%
Tough Strategies
Pointing Pairs				32563	77272	2.23%
Line/Box Reduction				15928	21826	0.63%
Gurths Theorem				0	0	0.00%
X-Wing				7686	8594	0.25%
Simple Colouring				17747	24202	0.70%
	Rule 2	6113	25.26%
	Rule 4	18089	74.74%
Y-Wing				14683	18510	0.53%
Rectangle Elimination				11747	16683	0.48%
Sword-Fish				846	860	0.02%
XYZ Wing				5405	6065	0.17%
Almost Locked Pair				0	0	0.00%
Bi-value Universal Grave				1069	1069	0.03%
X-Cycle				4605	5630	0.16%
	Strong Links	876	15.56%
	Weak Links	4545	80.73%
	Off-chain	209	3.71%
XY-Chain				16658	36159	1.04%
Diabolical Strategies
3D Medusa				5718	8012	0.23%
	Rule 1	473	5.90%
	Rule 2	226	2.82%
	Rule 3	689	8.60%
	Rule 4	2100	26.21%
	Rule 6	4200	52.42%
	Rule 7	324	4.04%
Jelly-Fish				10	10	0.00%
Unique Rectangle				2135	2285	0.07%
	Type 1	740	32.39%
	Type 2	158	6.91%
	Type 2b	46	2.01%
	Type 2c	11	0.48%
	Type 3	39	1.71%
	Type 3b	218	9.54%
	Type 4	661	28.93%
	Type 4b	236	10.33%
	Type 5	176	7.70%
Avoidable Rectangle				25	25	0.00%
Fireworks				111	112	0.00%
SK Loops				0	0	0.00%
Extended Unique Rectangle				101	102	0.00%
	Type 1	72	70.59%
	Type 2	0	0.00%
	Type 4	30	29.41%
Hidden Unique Rectangle				4911	6282	0.18%
	Type 1	3468	55.21%
	Type 2	1676	26.68%
	Type 2b	1138	18.12%
WXYZ Wing				3347	4116	0.12%
Aligned Pair Exclusion				3781	5379	0.15%
Extreme Strategies
Exocet				2	2	0.00%
	Rule 1	3	60.00%
	Rule 5	2	40.00%
Grouped X-Cycle				2529	3756	0.11%
	Strong Links	116	3.09%
	Weak Links	3498	93.13%
	Off-chain	142	3.78%
Finned X-Wing				0	0	0.00%
Finned Sword-Fish				387	396	0.01%
Franken Sword-Fish				0	0	0.00%
Alternating Infer. Chains				7995	24212	0.70%
	Strong Links	9351	38.62%
	Weak Links	12631	52.17%
	Off-chain	2230	9.21%
Sue-de-Coq				17	17	0.00%
Digit Forcing Chain				890	1899	0.05%
Nishio Forcing Chain				39	47	0.00%
Cell Forcing Chain				830	2274	0.07%
Unit Forcing Chain				208	553	0.02%
Almost Locked Sets				10	14	0.00%
Death Blossom				2	0	0.00%
Pattern Overlay				17	18	0.00%
Quad Forcing Chain				51	77	0.00%
Bowman Bingo				53	172	0.00%

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