HP00 Solving “hardest puzzles”




For players familiar with Sudoku Explainer, this thread is dedicated more or less to puzzles rating 10.5 and more, surely not to puzzles having a SE rating lower than 9.5.

I’ll nevertheless include puzzles with a lower rating having some of the key propertues described here.



Such puzzles can be solved using basic techniques plus AIC’s nets, especially if the set or rules includes “vicinity analysis” in ALS,AHS,AALS,AAHS but the path is generally  long and boring, nothing a player likes, although it remains for a computer an average very quick way to crack the puzzle


Other more sophisticated techniques have been experienced here and there to produce elegant non T&E solutions, to my view mainly in 2 directions:


AIC’s nets including other local patterns (eg: Sue de Coq)

AIC’s nets including AUR’s and all more complex forms of the same kind.


I will use in that chapter the simplest AUR.


Having worked a lot on all “hardest puzzles” published or kindly given by coloin, I could slowly split that data base in several families.


Others players of the forum brought some key pieces of solutions or fresh ideas to build that new view on “hardest” puzzles. It would be difficult to set up a full list, but I can not omit to mention


Steve KURZHALS who first found the well known SK loop.

Allan Barker, the main contributor to some of the key techniques described here after,

“Ronk” “ttt” ans “abi”who made very good things based on Allan findings

And the group of players having worked on the “symmetry” issue.


An attempt to make a “practical” typology in a data base  including more than 200 000 members (and much more in the future) is requiring a computer processing. As a matter of fact, the classification proposed here includes exclusively some solving techniques I could implement in my solver.


The last one I just coded after I tested several inefficient routines, is a generation of some specific big SLGs  derived form the Allan Barker Model.



Summary of common solving properties found in hardest puzzles


I will not try to define more precisely the hardest puzzles. What is for sure is that, when the difficulty to solve is clearly higher, the PM has some common specificities (sometimes after one or 2 singles have been found)


Up to now, I have identifies 4/5 key properties:


The following properties are not exclusive and will be explained later;


P1. SK loop                     a loop of 8 AAHS/AALS crossing 4 boxes

P2: EXOCET:      a base of 2 cells linked to a target of 2 other cells

                                 a small sub group is made of 2 interleaved EXOCETS .

P3. Multi floors fish:     Interleaved “quasi fish” patterns.

P4:EXOCET cousins  eg: Platinium Blonde  solving using AUR’s

P5. Symmetry                  Symmetry of given with several sub families



Subject to a final check, I have not found, up to now, a puzzle of the top list of hardest having none of these properties,



Relative Frequency of each property


I ‘ll try to give stats on the relative frequency of each, but theses stats are done on highly biased data, so the reality could be very different.


To limit the effect of all filters applied when searching “hardest” puzzles, I’ll use as far as I  can  the results of the last pattern games bringing some information related to that discussion, but, here again, we start from specific pattern having, among others, a symmetry constraint.


As of to day, the most common property seems to be the EXOCET pattern,

The next one should be the SK loop, but I was lacking a high speed process to detect puzzles having the multi floors fish property.


Symmetry is and will remain very confidential.



P1. The SK loop


The SK loop has been first seen by Steve KURZHALS. in Easter Monster a puzzle from JPF

I show it in the way my solver handles it, slightly different from the initial findings of Steve.


The PM at the start of Easter Monster is the following


1    478  3458 |3567 389 5678 |3489 369  2     

238  9    378  |4    126 1267 |138  5    368 

3458 248  6    |1235 389 1258 |7    139  3489


2468 5    1478 |9    126 3    |128  1267 678   

389  126  389  |126  7   4    |3589 126  3589

2369 1267 1379 |8    5   126  |1239 4    3679


7    148  4589 |1235 348 1258 |6    239  3459 

456  3    145  |1267 126 9    |245  8    457 

4589 468  2    |3567 348 5678 |3459 379  1  


The SK loop forms a square with in red the 4 intersections


r2c13 r13c2 r79c2 r8c13 r8c79  r79c8 r13c8 r2c79 =>loop

3827  2748  4816  1645  4527   2739  3916  1638


In that  sequence each AAHS/AALS sahres 2 digits with  the next one.


The SK loop is described at several places in my site, partly as a closed virus chain in the tagging process  level 4 , and in the first sample  here where one can find a detailed description of effects



The pattern must be identical to that one:


. 4 boxes with 4 empty corners forming a square/rectangle 2 rows,2 columns

. In each box,  2 AAHS/AALS  of {2 cells 4 digits} each, one row, one column

. Shared digits can be chained without overlapping forming a loop


Other kinds of loops of the same family could exist on the paper (six boxes for example), but no example has been seen.


SK loop is by far not the most common property in the “hardest” data base. It can be found in “relatively” easier puzzles.


Some patterns have a high potential in puzzles having the SK loop. Recently, competing in the game 112, I generated thousands puzzles having the SK loop with that pattern. I’ll write a special chapter on that game;