Since i am rather busy these weeks, just a quick summary, what i found out about these properties, which can be calculated with dobrichev's program.
Thanks for the permission to use it. It is very fast for what it does.
For others, which use it: Be careful, that with the --unav --minus1 option the puzzle is minimal, otherwise the program crashes. If you compile it yourself, this is easy to fix.
Replace
int maxUaSizeForClue = g.usetsBySize.rbegin()->nbits;
by
int maxUaSizeForClue = 0;
if (g.usetsBySize.size() > 0) maxUaSizeForClue = g.usetsBySize.rbegin()->nbits;
To compile it under linux, i had to make some changes, which hopefully dont affect the results.
Let me call it a dob-zero puzzle, when you can't derive any number, as soon as one of the givens is dropped.
I looked, how much of the puzzles with high ER had this property.
Clearly, the higher rated the known toughies are, the higher is the percentage of dob-0 puzzles.
For these dob-0 puzzles also the minimum and average sizes of the maxUA values are the higher, the higher the ER is (maxUA is the maximum size of the unavoidable sets you get, when a number is dropped from the givens).
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#puzzles #dob-0 perc. min maxUA avg maxUA
ER 11.9 5 3 60 % 46.0 51.1
ER 11.5+ 271 52 19 % 46.2 50.7
ER 11+ 8666 692 8 % 45.6 50.2
ER 10.6-10.9 18772 870 4.6% 45.2 49.9
For non dob-0 puzzles these maxUA values are of minor interest for me. Often it is simple to find one or more numbers, in this case you would get different UA sizes after adding them.
I tried to modify the propram in the way that i added clues, which can be derived with n-1 givens. So you get a non mimimal puzzle. Removing a non mimimal clue will always allow to solve the whole puzzle. So i continued the procedure with dropping the minimal clues only and looked, if i came to a puzzle, where no clues could be added. However this happened all the time, because at some point no non minimal clues remained.
Nevertheless some non minimal puzzles with the dob-0 property are high rated (including Mauricio's 37). But compared to the minimal ones they are more common under puzzles with lower ER's.
On the other hand I tested some 100 dob-0 puzzles out of my step1 sets with 21-23 clues. The average ER of the dob-0's was always below 10.5 (in my search i had stopped to rate puzzles sets, when the avg ER was below 10.6). Also i found higher maxUA sizes in ER 10.2 puzzles than in the 11.9's.
The lower the number of clues, the more common are dob-0 puzzles in my step1 sets. This is, what i have so far:
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# clues #puzzles dob-0 perc. expected #dob-0
21 503367 22482 4.47 % 22482
22 1414631 10605 0.75 % 33573
23 6853073 2043 0.03 % 2233
So there will be more than 50000 dob-0 puzzles in my sets.
When i combined the dob-0 property with my 2 fast filters, the resulting puzzle sets gave an average ER around 10.6. This way i had found The "Second Flush" puzzle very soon (and luckily, because the next best rated puzzle had 11.5).
Probably in a week it should be finished to calculate the dob-0's in my sets and rate those, which also pass my 2 filters.
