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The Relative Incidence of Sudoku Strategies

From sudokuwiki.org, the puzzle solver's site
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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
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 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:


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.

STRATEGYCount%
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%
Sword-Fish835 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%
Jelly-Fish8 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 Sword-Fish368 0.17%
Franken Sword-Fish1 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