The New Sudoku Players' Forum

by **coloin** » Wed Apr 19, 2006 10:29 pm

Suffice to say - Its fairly difficult to generate these minimal puzzles with more than 30 clues !

If anyone wants a go - dukuso's program http://magictour.free.fr/suex9s.exe will generate them from a text string.

C:\sudoku\suex9s [file.txt] seed 100000 28 - this will generate several minimal sudokus with more than 28 clues from a text file eg the SF grid

Code: Select all: 639241785284765193517983624123857946796432851458619237342178569861594372975326418

More clues are generated if you selectively remove non essential clues from the string. eg remove one box and all of one number:

Code: Select all: 63.2417852847651.3517.83624123....467.6...851458...23734217856.8615.4372.75326418

This way I generated 6 more 34s in the "SF" grid

Code: Select all: SF grid 6.9.4..8..84.65..35..9.36..........679..3.85...8..923..4...8...8...9437..7.32.4.. 6.9.4..8..84.65..35..9.36..........679..3.85...8..923..4...8.......9437..7.32.4.8 6.9.4..8...4.65..35..9836..........679..3.85...8..923..4...8...8...9437..7.32.4.. 6.9.4.....84.65..35..9836..........679..3.85...8..923..4...8.......9437..7.32.4.8 6.9.4.....8..65..35.79836..........679..3.85......923..4...8...8...9437...53264.. 6.9.4.....8..65..35.79.36.....8....679..3285......923..4...8...8...9.37...53264.. 6.9.4......4.65..35..9836..........679..3.85...8..923..4...8...8...9437..7532.4.. 6.9.4......4.65..35..9836..........679..3.85...8..923..4...8...8...9437..7.32.4.8 6.9.4........65..35.79.36.....8....679..3285...8..923..4...8...8...9.37...53264..

And two 34s in the "SFB" grid

Code: Select all: SFB grid 5.9...61..21...37974.....25..........1756..9329..4.15.96..7......26.....1.42.3.6. 5.9...61..21...37974.....25..........1756..9329..4.15.9...7..3...26.....1.42.3.6.

Just how to find 35s in these grids ? - the chances are they are there !
C

by **Ocean** » Wed Apr 26, 2006 12:44 am

coloin wrote:... eg the SF grid
Code: Select all
639241785284765193517983624123857946796432851458619237342178569861594372975326418

More clues are generated if you selectively remove non essential clues from the string. eg remove one box and all of one number:
Code: Select all
63.2417852847651.3517.83624123....467.6...851458...23734217856.8615.4372.75326418

This way I generated 6 more 34s in the "SF" grids.
Code: Select all
SF grid 6.9.4..8..84.65..35..9.36..........679..3.85...8..923..4...8...8...9437..7.32.4.. 6.9.4..8..84.65..35..9.36..........679..3.85...8..923..4...8.......9437..7.32.4.8 6.9.4..8...4.65..35..9836..........679..3.85...8..923..4...8...8...9437..7.32.4.. 6.9.4.....84.65..35..9836..........679..3.85...8..923..4...8.......9437..7.32.4.8 6.9.4.....8..65..35.79836..........679..3.85......923..4...8...8...9437...53264.. 6.9.4.....8..65..35.79.36.....8....679..3285......923..4...8...8...9.37...53264.. 6.9.4......4.65..35..9836..........679..3.85...8..923..4...8...8...9437..7532.4.. 6.9.4......4.65..35..9836..........679..3.85...8..923..4...8...8...9437..7.32.4.8 6.9.4........65..35.79.36.....8....679..3285...8..923..4...8...8...9.37...53264..

C

You have got a fine nose for finding prospective grids.
In case more seeds are needed: here are two extra minimal 34s in the SF-grid.

Code: Select all: 6.9.4.....8..65..35.79836..........6.9..3.85......923..4..78...8....437..753264.. 6.9.4........65..35.79836..........6.9..3.85...8..923..4..78...8....437..753264..

by **coloin** » Thu Apr 27, 2006 1:46 pm

Ah....... two more 34s !

I thought I had done well....but now I'm not so sure.

I took a completly random grid, I "randomly but educatedly" reduced it, and selectively found higher and higher minimal clues.

Code: Select all: +---+---+---+ |126|347|598| |458|169|732| |379|285|461| +---+---+---+ |213|478|659| |584|692|317| |697|513|824| +---+---+---+ |732|851|946| |845|926|173| |961|734|285| +---+---+---+

Until I got 33s ! - either I struck lucky [unlikely] - or I have unknowingly improved my method.

Code: Select all: ....47.9.4.8.69.32..92.....2.3478.595...92..7.9...38...32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2.3478.595...9...7.9...382..32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2.347..5958..92..7.9...38...32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2.347..5958..9...7.9...382..32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2..478.595...923.7.9...38...32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2..478.595...9.3.7.9...382..32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2..47..5958..923.7.9...38...32....4.84.....73......2.. ....47.9.4.8.69.32..92.....2..47..5958..9.3.7.9...382..32....4.84.....73......2.. ....47.9...8.69732..92.....2.3478.595...92..7.9...38...32....4.84.....73......2.. ....47.9...8.69732..92.....2.3478.595...9...7.9...382..32....4.84.....73......2.. ....47.9...8.69732..92.....2.347..5958..92..7.9...38...32....4.84.....73......2.. ....47.9...8.69732..92.....2.347..5958..9...7.9...382..32....4.84.....73......2.. ....47.9...8.69732..92.....2..478.595...923.7.9...38...32....4.84.....73......2.. ....47.9...8.69732..92.....2..478.595...9.3.7.9...382..32....4.84.....73......2.. ....47.9...8.69732..92.....2..47..5958..923.7.9...38...32....4.84.....73......2.. ....47.9...8.69732..92.....2..47..5958..9.3.7.9...382..32....4.84.....73......2.. ....4..9...8.69732.792..4..2..47..5958..92..7.9...38...32....4.84.....73......2.. ....4..9...8.69732.792..4..2..47..5958..9...7.9...382..32....4.84.....73......2.. ....4..9...8.69732.792.....2.347..5958..92..7.9...38...32....4.84.....73......2.. ....4..9...8.69732.792.....2.347..5958..9...7.9...382..32....4.84.....73......2.. ....4..9...8.69732.792.....2..47..5958..923.7.9...38...32....4.84.....73......2.. ....4..9...8.69732.792.....2..47..5958..9.3.7.9...382..32....4.84.....73......2.. ....4..9...8.69732.7.2.....2.3.7..5958..92..7.97..38.4.32....4.84......3......28. ....4..9...8.69732.7.2.....2.3.7..5958..9...7.97..3824.32....4.84......3......28. ....4..9...8.69732.7.2.....2...7..59584.92..7.97..38.4.32....4.84......3......28. ....4..9...8.69732.7.2.....2...7..59584.9...7.97..3824.32....4.84......3......28. ....4..9...8.69732.7.2.....2...7..5958..923.7.97..38.4.32....4.84......3......28. ....4..9...8.69732.7.2.....2...7..5958..9.3.7.97..3824.32....4.84......3......28.

And here is a more distant one

Code: Select all: .2...7.9.458.697.2.79.8.4..........9584.923.7.97..38...3.....4.84..2........3.2..

Maybe those 34s dont look quite so good now.

by **coloin** » Tue May 09, 2006 10:19 pm

I tried to get more 34s in the SF - but not successful .......HOWEVER

coloin wrote:Maybe those 34s dont look quite so good now.

Taking the grid that I found 33s in - the previous post

Code: Select all: +---+---+---+ |126|347|598| |458|169|732| |379|285|461| +---+---+---+ |213|478|659| |584|692|317| |697|513|824| +---+---+---+ |732|851|946| |845|926|173| |961|734|285| +---+---+---+

This grid was constructed at random - It is in a posting from here - http://forum.enjoysudoku.com/viewtopic.php?t=3177&postdays=0&postorder=asc&start=15

Amalgamating the 33s this 34 popped out !

Code: Select all: .2..4..98.58..97.2.7....4....347...9584.923.7.9...382..3.......84..26..3......28.

More and more 34s were generated...untill I got a minimal 35

Code: Select all: +---+---+---+ |.26|.4.|.98| |458|.69|7.2| |.7.|...|...| +---+---+---+ |..3|47.|..9| |584|.92|3.7| |.9.|..3|82.| +---+---+---+ |.3.|...|...| |84.|.26|..3| |...|...|.8.| +---+---+---+

The "coloinmax" - and here is its partner.

Code: Select all: .26.4..98458.697.2.7.........347...9584.923.7.9...382..3.......84..26..3.......8. .26.4..98458.697.2.7.........347...958..923.7.9...3824.3.......84..26..3.......8.

I think Ocean commented before that these puzzles are difficult - I cant go beyond

Code: Select all: +---+---+---+ |.26|.4.|.98| |458|.69|7.2| |.7.|28.|...| +---+---+---+ |2.3|478|..9| |584|.92|3.7| |.9.|..3|824| +---+---+---+ |.3.|8..|...| |84.|.26|..3| |...|.3.|.8.| +---+---+---+

by **Ocean** » Wed May 10, 2006 12:05 pm

coloin wrote:More and more 34s were generated...untill I got a minimal 35
Code: Select all
+---+---+---+ |.26|.4.|.98| |458|.69|7.2| |.7.|...|...| +---+---+---+ |..3|47.|..9| |584|.92|3.7| |.9.|..3|82.| +---+---+---+ |.3.|...|...| |84.|.26|..3| |...|...|.8.| +---+---+---+

The "coloinmax" - and here is its partner.

Code: Select all
.26.4..98458.697.2.7.........347...9584.923.7.9...382..3.......84..26..3.......8. .26.4..98458.697.2.7.........347...958..923.7.9...3824.3.......84..26..3.......8.

Good job with that one, coloin!
Now we have these questions:
1. How many grids have minimal 35s?
2. Does any grid contain a minimal 36 (or higher) ?

by **coloin** » Wed May 10, 2006 2:11 pm

Thanks indeed........I think you are the only one around who can truly appreciate the 35 !

I took a random grid to see how far I could get....and it had a 35..

Is it almost certain that at least 1 in 20 grids have a 35 ? [a very casual statistical reverse from p<0.05 !]. It probably is more than this.

As the number of clues goes up the number of options for clues increase

Code: Select all: 34 clues 81!/34!47! = 0.75*10^23 35 clues 81!/35!46! = 1.01*10^23 36 clues 81!/36!45! = 1.30*10^23 37 clues 81!/37!44! = 1.58*10^23

but space for one more unavoidable set is reduced by one.

A 36 is going to be very difficult to find.....maybe the fact that we now know there must be more 35s around will encourage us.

I have a general hunch that the grids with 17s have a greater number of minimal puzzles - and therefore these are the ones which may have 35s. I dont know what the minimum for my grid is.....it is unlikely to have a 17 so that does not particularly back that up. [checker 17 running !]

Thanks for your support !

I will try another grid meanwhile...any suggestions ?

Is there anybody else out there who is up to the challenge ?

The programs I use are available from here

http://magictour.free.fr/suex9s.exe
http://magictour.free.fr/suexmu35.exe
http://magictour.free.fr/suexsfl.exe
http://magictour.free.fr/suexmult.exe

I will give an example of a command line to get the most computor green [like me] to start producing puzzles - if anyone wants.

Here
Make a directory eg sudoku
Copy suex9s.exe to this directory
Copy this text file [open notepad and paste the string of numbers - call it file1.txt] to the directory "sudoku"
text string is

Code: Select all: ....47.9.4.8.69.32.792..4..2.3478.59584.923.7.9...3824.32....4.84.....73......28.

Click on Programs - Asessories - Command prompt
Type cd c:\
Type cd sudoku
Type suex9s file1.txt seed 10000 31
Hit enter

[This generates puzzles greater than 31 clues from 10000 attempts]

There is nothing better than hearing that processor whine and watching the screen come up with reams of grids !

Roll on the 36

C

by **JPF** » Wed May 10, 2006 10:40 pm

coloin wrote:Is there anybody else out there who is up to the challenge ?

Thanks for the explanations how to use http://magictour.free.fr/suex9s.exe ...
I'm not sure I understand what I did, but I dit it.
I got 31 33 clues-minimal puzzles...

but where this puzzle

Code: Select all: ....47.9.4.8.69.32.792..4..2.3478.59584.923.7.9...3824.32....4.84.....73......28.

is coming from ?

anyway, I will try again.

JPF

by **coloin** » Thu May 11, 2006 4:02 pm

Ah I am glad you managed it.....I hope others have as well.....

The example file was taken from the grids from 4 posts ago - it is an amalgum of all the 33s I found prior to the finding of 34s.

The file was to demonstrate generating puzzles - many more will be generated if you set the clue number to more than 30 !

It also shows how easy it is to produce 33s....if only !

If you put a random file in you only tend to get 28s or 29s at most - and that is generating [seed value] 100000 puzzles.

Chosing the clues to leave out is the tricky one - you can start easily by removing all of one clue number and one complete box.

A random Grid [Ran1.txt] and two reduced forms

Code: Select all: 123749568456183927789625431238567149561498273947231685874952316615374892392816754 .23749568456.8392778962543.238567.4956.49827394723.6858749523.66.53748923928.6754 ...749568....83927...62543.238567.4956.49827394723.6858749523.66.53748923928.6754

Run these with a seed of 27 or 28 or 29 to see the difference
suex9s ran1.txt seed 10000 27
followed by 10000 28 etc [do all three grids at the same time]
Only the second and third form gives many grids larger than 28 at the seed 10000. The higher the seed the more you will get - although it will take longer !

It is a matter of generating step wise larger puzzles witha mixture of random generation - using suex9 - and full enumeration using suexmu35.

A longer post will follow shortly - I am recording my grids for a new post - see how far I get. Some of the stages take a while, sometimes an hour !

I must at this stage acknowlege the work done bu Guenter [dukoso] - in providing me with all the programs.

Regards C

by **JPF** » Fri May 12, 2006 12:30 am

Reading the recent posts of this thread, it came to me some side remarks.
Hope I'm not too far from the subject.

This 35 clues puzzle from Ocean is minimal :

Code: Select all: 1 . . | . 5 6 | 7 8 . 7 8 . | 1 2 . | 4 5 . 4 . . | 7 . . | 1 . 3 -------+-------+------- 2 . 1 | . 6 4 | 8 . . . 6 . | . 9 . | . . 1 8 . . | 2 . . | 5 . . -------+-------+------- 3 . 2 | 6 4 5 | . . . 6 4 . | . 7 . | . . . . . . | . . 2 | . . .

When we add one valid clue c to this puzzle P, we get a puzzle P’=P+c non-minimal (by definition)
Let n(X) be the clue number of a puzzle X ; n(P’)=36.

Let’s take c : (98)=(r9c8)=[4].
Then we can extract from P’ several minimal puzzles.
P is obviously one of them.

Here is Pm, the one with the minimum number of clues n(Pm)=28 :

Code: Select all: * . . | . 5 6 | 7 8 . 7 8 . | 1 * . | * * . * . . | 7 . . | 1 . 3 -------+-------+------- * . 1 | . 6 4 | 8 . . . 6 . | . 9 . | . . 1 8 . . | * . . | 5 . . -------+-------+------- 3 . 2 | 6 * 5 | . . . 6 4 . | . 7 . | . . . . . . | . . 2 | . 4 .

* indicates the removed clues.

My first remark is that by adding one (valid) clue to a n-minimal puzzle, one gets a puzzle which can contain minimal puzzles with a number of clues n' well below n ; (up to n-7 in this puzzle)
The cell 98(r9c8) is the only one providing this gap.
For the others : 0<=n-n'<7

On the other hand, if we add to the initial puzzle P the cell c :(14)=[4], there is no minimal puzzle included in P+c except P itself.
n'-n=0
The cell 14 is the only one having this property.

And finally a general question by extending the first remark :

M : is a minimal puzzle
S : is the grid solution of M.
c : a cell (including the digit) of S not included in M

The puzzle P=M+c is valid, but not minimal.

It exists at least one valid minimal puzzle X such that X<M+c.
(Obviously, M is one of them).

What is the maximum value for v = n(M)-n(X) ; for all the possible values of X ?

For the Ocean's 35-minimal puzzle above, it seems that v=7
For the 17s , v=0.

Today, in any case the upper limit of v is 35-17=18.

But probably, all this has nothing to do with the search of the max number of clues of the minimal puzzles.

JPF

by **ronk** » Fri May 12, 2006 1:10 am

JPF wrote:It exists at least one valid minimal puzzle X such that X<M+c.
(Obviously, M is one of them).

Wouldn't that be ... X<=M+c?

JPF wrote:What is the maximum value for v = n(M)-n(X) ; for all the possible values of X ?

For the Ocean's 35-minimal puzzle above, it seems that v=7
For the 17s , v=0.

Today, in any case the upper limit of v is 35-17=18.

Which is not the same as saying the upper limit of 18 is possible for a minimal 35, is it?

by **JPF** » Fri May 12, 2006 7:54 am

ronk wrote:
JPF wrote:It exists at least one valid minimal puzzle X such that X<M+c.
(Obviously, M is one of them).

Wouldn't that be ... X<=M+c?

We may keep the strict inclusion, because M+c is not minimal whereas X is minimal. So X#M+c
and therefore X<=M+c => X<M+c.

ronk wrote:
JPF wrote:What is the maximum value for v = n(M)-n(X) ; for all the possible values of X ?

For the Ocean's 35-minimal puzzle above, it seems that v=7
For the 17s , v=0.

Today, in any case the upper limit of v is 35-17=18.

Which is not the same as saying the upper limit of 18 is possible for a minimal 35, is it?

The smallest minimal puzzle X included in a puzzle M has at least 17 clues.
so : n(X)>=17
if the minimal puzzles are such that n(M)<=35, we can therefore say that n(M)-n(X)<=35-17=18 ==> v<=18

But it's probably not true that every M puzzle has a 17 in it.
And I don't know if we can find a 17 in a 35 minimal puzzle.

JPF

by **coloin** » Sat Jun 03, 2006 8:58 pm

This Random grid - [ran1.txt] from a few posts back

Code: Select all: 123749568456183927789625431238567149561498273947231685874952316615374892392816754

Inexplicably I have found four minimal 35 s

by **coloin** » Fri Jun 09, 2006 11:39 pm

Here is the first random grid from the list

Code: Select all: Ran1 - 123749568456183927789625431238567149561498273947231685874952316615374892392816754

A 35 was eventually found

Code: Select all: +---+---+---+ |..3|..9|...| |4..|...|...| |7.9|6..|.3.| +---+---+---+ |..8|.6.|.4.| |5..|...|..3| |9.7|2..|685| +---+---+---+ |8.4|.52|3.6| |6.5|3.4|89.| |.92|8..|.54| +---+---+---+

Set of 4 35s

Code: Select all: ..3..9...4........7.96...3...8.6..4.5.......39.72..6858.4.523.66.53.489..928...54 ..3..9...4........7.96...3...8.6..4.5.......39.72..6858.4.52..66.53.489.3928...54 ..3..9...4........7.96...3...8....4.5.......39.72..6858.4.523.66.53.489..928.6.54 ..3..9...4........7.96...3...8....4.5.......39.72..6858.4.52..66.53.489.3928.6.54

Along with the grids previously known to have a 35, I have looked furthur at these two grids - each at opposite ends of a grid characteristic.
SFB grid - no 4-set unavoidables - low MCN - low average minimal grid count
dukuso15 - an incredible 36 4-set unavoidables - high MCN - high average minimal grid count

Code: Select all: Ran 1 - 123749568456183927789625431238567149561498273947231685874952316615374892392816754 Dukuso15 - 123568479864791352957243681218657934536489127749312865391825746472136598685974213 SFB - 589732614621854379743916825835129746417568293296347158968471532372695481154283967 Coloinmax- 126347598458169732379285461213478659584692317697513824732851946845926173961734285 Canon - 123456789789123456456789123231564897564897231897231564312645978645978312978312645 RW - 123469785456187932789253164371694528698512347542378691215946873864735219937821456

Code: Select all: Ran 1 Dukuso15 SFB Coloinmax Canon SF RW Number of 28 clue puz per mill 2372 9744 1378 4084 50751 1554 Number of 29 clue puz per mill 145 956 83 271 6735 95 Number of 30 clue puz per mill 5 61 3 10 466 1 Number of 31 clue puz per mill 0 4 1 0 20 0 Min. number of clues 19 20 17 19 20 17 20 Number of 35s found 4 0 0 2 1296 0 2 Number of 34s found - 3888! 80 - - 42 -

The clue counts do not appear to be too relevant - and they are skewed in favour of small puzzles. I think there are just so many puzzles that we are looking at the "tip of an iceberg" when we generate clue count statistics.

As I have specifically looked at only one region of the above grids - there may be many 34 clue regions in the above grids.
As more clues are included the number of 34s increases. - The number of 34s in the dukuso15 coincided with a development of a more efficient search.
The number of 34s in the SFB went up from 2 to 80 !
I dare to suggest that many many grids have 34s in them - if not 35s. The problem is deciding which 34s have 35s lurking near.

by **coloin** » Wed Nov 01, 2006 8:32 pm

An update !

These are the known 35 clue minimal puzzles

Code: Select all: MC 1...5678.78.12.45.4..7..1.32.1.648...6..9...18..2..5..3.2645...64..7.........2... 1.3..678.78.12.45.4..7..1.32.1.648...6..9...18..2..5..3.2645...64..7.........2... Coloinmax .26.4..98458.697.2.7.........347...9584.923.7.9...382..3.......84..26..3.......8. .26.4..98458.697.2.7.........347...958..923.7.9...3824.3.......84..26..3.......8. Ran1 ..3..9...4........7.96...3...8.6..4.5.......39.72..6858.4.523.66.53.489..928...54 ..3..9...4........7.96...3...8.6..4.5.......39.72..6858.4.52..66.53.489.3928...54 ..3..9...4........7.96...3...8....4.5.......39.72..6858.4.523.66.53.489..928.6.54 ..3..9...4........7.96...3...8....4.5.......39.72..6858.4.52..66.53.489.3928.6.54 RW 1..4..7.545.1..93.78..53..4.7..9.5...98.1..4.5..3.8.9..1..4687386..3.........1... 1..4..7..45.1..93.78..53..4.7..9.....98.1.34.5..3.8.9..1..4687386..3.........1.5.

The Dukuso15 grid intrigued me - I was unable to find a 35 despite over 4000 34s ! Analysis by JPF revealed many chameleon puzzles - every clue was mutable / each clue had 2 solution grids. see

coloin wrote:The problem is deciding which 34s have 35s lurking near.

I searched around this 34 clue puzzle with the highest suexrat rating from the collection of 34s

Code: Select all: rating: 106 , ..........6...1352.5..436.1.1..579.....4.9..7..9.1..6...1..57.6.721..5...85.742.3

- and I did find 35s !

Code: Select all: Duk15 rating: 85 , ..........64..135..57.436.1.1..5793.5....9..7...31..65..1..57.6.7213.....8..742.. rating: 84 , ..........64..135..57.436.1.1..5793.5....9..7...31..6...1..57.6.7213.5...8..742.. rating: 83 , ..........64..135..57.436.1.1..5793......9..7...31..65..1..57.6.7213.....85.742.. rating: 87 , ..........64..135..57.436.1.1..5793......9..7...31..6...1..57.6.7213.5...85.742.. rating: 81 , ..........64..135..57.4.6.1.18.5793.5....9..7..931..65..1..57...7.13.....8..742.3 rating: 84 , ..........64..135..57.4.6.1.18.5793.5....9..7..931..6...1..57...7.13.5...8..742.3 rating: 84 , ..........64..135..57.4.6.1.18.5793......9..7..931..65..1..57...7.13.....85.742.3 rating: 85 , ..........64..135..57.4.6.1.18.5793......9..7..931..6...1..57...7.13.5...85.742.3 rating: 46 , ..........6...1352.57...6.1.18.5793....4.9..7..931..6...1..57...7.13.5...85.742.3 rating: 46 , ..........6...135..57..36.1.1..5793.5.64.9.....9.1..6...1.257...7213.5...8..742.3 rating: 66 , ..........6...135..57...6.1.1..5793....4.9..7..931..6...1.257.6.7213.5...85.742.. rating: 46 , ..........6...135..5..436.1.1..5793.5.64.9.....9.12....91.257...7213.5......742.3 rating: 46 , ..........6...135..5..436.1.1..5793.5.64.9.....9.12.....1.257...7213.5...8..742.3

by **JPF** » Fri Nov 03, 2006 8:35 pm

Glad you finally got 35s in this grid.

All these puzzles are not easy to solve (7.1≤ ER ≤7.8) except 4 singles (suexrate=46).

coloin wrote:I searched around this 34 clue puzzle with the highest suexrat rating from the collection of 34s

Why did you pick the highest rated 34 ?

JPF

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