SudokuWiki.org
Strategies for Popular Number Puzzles
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.
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:
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.
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.
*
*
Note: The first column is how many puzzles where the strategy is used. The second is how many instances were found overall.
The % is the number of times that strategy is used compared to the sum of all strategies.
The % inside specific strategies (eg UR) is the % within that strategy and should add up to 100%
The % is the number of times that strategy is used compared to the sum of all strategies.
The % inside specific strategies (eg UR) is the % within that strategy and should add up to 100%
| 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
The answer to the reader's original question - the incidence of 'evil' strategies, is I'd say, about 5%.
Andrew Stuart
Comments
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... by: Manohar Kulkarni
Two questions:
(1) The post dies not contain data on the % of tough puzzles. Can you please furnish?
(2) Which are the, "Evil" strategies? Are they included in the grade, "Extreme"/
... by: poppinfresh
Since I highly doubt that all possible solutions paths are followed for a single puzzle, I then guess what happens is that, the topmost (easiest) strategy is CHOSEN to be performed to get to the next step of puzzle, though ALL possible moves for that step are counted??? Not sure if the unused strategies get recounted (for calculating the percentages) if the move is still available on the next step?
Perhaps there is simply just a bug in the code that calculated the percentages. Either way, I am extremely curious how to interpret these numbers!
... by: Uhm
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
... by: Fred
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 ?
... by: slowofmind
... by: PeteTy
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.
... by: keith
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...
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
... by: Smashblast
... by: joseph brophy