## Number of minimal puzzles per ED valid pattern with N clues

Everything about Sudoku that doesn't fit in one of the other sections

### Re: Number of minimal puzzles per ED valid pattern with N cl

Yes that program works well !

so to estimate the number of puzzles in a 25 clue pattern
average number of puzzles was 1830 in 40 ed grids.
1830/40 * 5e9 = which gives an average count of 2.2e11 25-puzzles per pattern.

of note the asymmetric diagonal 25 clue pattern had 21621 puzzles in the 40 grids - roughly 10 times more puzzles ....
Code: Select all
`+---+---+---+|..5|..8|..9||...|.2.|.7.||6..|1..|3..|+---+---+---+|..6|8..|5..||.9.|.6.|..1||7..|..5|.4.|+---+---+---+|..1|3..|.8.||.5.|.4.|7..||2..|..1|...|+---+---+---+ 25 clues - 21621 puzzles`

Code: Select all
`+---+---+---+|1..|2..|...||.3.|..4|...||..5|.6.|4..|+---+---+---+|4..|...|7..||..1|.5.|..2||.2.|..3|.6.|+---+---+---+|..3|7..|8.6||...|..8|.47||...|.1.|53.|+---+---+---+ 25 clues - patterns game 0269 - 1993 puzzles `

C
coloin

Posts: 1733
Joined: 05 May 2005

m_b_metcalf (2017.Feb.1) wrote:

This pattern is related to 258,
which reached 11.3,
but has a mini-diagonal less.

b9 disturbed,

does this make fewer puzzles ?

OR, with "same" number of puzzles,
does it make them easier ??

Pat

Posts: 3641
Joined: 18 July 2005

### Re: Number of minimal puzzles per ED valid pattern with N cl

For patterns with 23 clues
On average taking a few random patterns .... there are 1.5 puzzles per ed grid - this gives 7.5 e9 23-clue minimal puzzles per pattern

for pattern 258 there were 2.9 puzzles per ed grid on average
for pattern 280 there were 2.5 puzzles per ed grid on average
Code: Select all
`258              280          +---+---+---+    +---+---+---+|1..|2..|..3|    |9..|7..|..3||.4.|.5.|.2.|    |.4.|.5.|.9.||..3|...|1..|    |..2|...|6..|+---+---+---+    +---+---+---+|4..|..3|.6.|    |5..|..3|.1.||.1.|.7.|5..|    |.6.|.2.|4..||...|6..|...|    |...|9..|...|+---+---+---+    +---+---+---+|..8|.6.|7..|    |..5|.4.|...||.3.|4..|.8.|    |.8.|1..|.34||2..|...|..9|    |7..|...|.5.|+---+---+---+    +---+---+---+`
coloin

Posts: 1733
Joined: 05 May 2005

### Re: Number of minimal puzzles per ED valid pattern with N cl

My count for pattern 280 is 2.24.
Are you taking into account the minimality?
dobrichev
2016 Supporter

Posts: 1615
Joined: 24 May 2010

### Re: Number of minimal puzzles per ED valid pattern with N cl

Not sure ...they look all minimal .... i checked it with 400 grids x2 , maybe not enough
Of note the figure for a few random patterns [23 clues] grids is significantly larger than predicted at start of the thread .... 1.1e9 vv 7.5 e9
I can only presume the varience between patterns is significant ...

its logical that if there are more puzzles - then there is increased likelihood of a high rated puzzle - SE > 11
Pattern 258 did have more puzzles and also more higher rated puzzles .
coloin

Posts: 1733
Joined: 05 May 2005

### Re: Number of minimal puzzles per ED valid pattern with N cl

I looked at a few more patterns
It seems there is quite a spread

6.9e12 - number of 23 clue minimal puzzles per grid
3.2e13 - number of ed patterns with 23 clues
This gives 1.07 e9 puzzles per 23 clue pattern ....

there should be 8 minimal puzzles with that pattern in 40 grids on average - These results don't really match !
Code: Select all
`24........3..9.85....8..........2.491.2..7......6..5......5..9...72...8.8.3.....6  - 540 puzzles in 40 grids                     3....1.5.69.....8.....5.....1...23.4.....7....6.3..1......7.5..2......3.5.7.4...2  - 355 puzzles in 40 grids                     1.6.........48..1..7.2..5....4.3....3...4..97..59......6........1.5..63....7....2  -  72 puzzles in 40 grids                     ....3.....2...1.6.1...2...........3.8..34.1.59.5.....85.....87..4.1........5.6..9  -  48 puzzles in 40 grids                     .......9..21...8.6..3.........1.4......2..7.949.....6...27......4..58..1...3..98.  -  38 puzzles in 40 grids  [empty box]        ...6.....7...9...........41...1..5..3...89.....4.2..6.1....2.34...3.4.9...5.1.8..  -   8 puzzles in 40 grids  [5 clues in band 1]6...........2...............1...4.6.3.9...7....6.75..9.48.9...5.....7.9..27.6.31.  -   0 puzzles in 40 grids  [2 clues in band 1]`
coloin

Posts: 1733
Joined: 05 May 2005

### Re: Number of minimal puzzles per ED valid pattern with N cl

At the rightmost are my counts per these 100 random grids.
Code: Select all
`24........3..9.85....8..........2.491.2..7......6..5......5..9...72...8.8.3.....6  - 540 puzzles in 40 grids                      (1096/100)3....1.5.69.....8.....5.....1...23.4.....7....6.3..1......7.5..2......3.5.7.4...2  - 355 puzzles in 40 grids                      ( 687/100)1.6.........48..1..7.2..5....4.3....3...4..97..59......6........1.5..63....7....2  -  72 puzzles in 40 grids                      ( 151/100)....3.....2...1.6.1...2...........3.8..34.1.59.5.....85.....87..4.1........5.6..9  -  48 puzzles in 40 grids                      ( 107/100).......9..21...8.6..3.........1.4......2..7.949.....6...27......4..58..1...3..98.  -  38 puzzles in 40 grids  [empty box]         (  72/100)...6.....7...9...........41...1..5..3...89.....4.2..6.1....2.34...3.4.9...5.1.8..  -   8 puzzles in 40 grids  [5 clues in band 1] (  19/100)6...........2...............1...4.6.3.9...7....6.75..9.48.9...5.....7.9..27.6.31.  -   0 puzzles in 40 grids  [2 clues in band 1] (   0/100)`

The proportions are close enough.
dobrichev
2016 Supporter

Posts: 1615
Joined: 24 May 2010

Code: Select all
` 280    25810.0   10.7   highest "diamond"10.6   11.3   highest "pearl"`

the two games differ by 0.7
explained by the difference in number of puzzles ??

Pat

Posts: 3641
Joined: 18 July 2005

### Re: Number of minimal puzzles per ED valid pattern with N cl

to Pat
It has to do with the number of puzzles - but also the percentage of difficult puzzles ...
for example
you could probably do better but - after three cycles the best i could do with the profligate pattern was
Code: Select all
`35........1..2.57....9..........3.968.1..7......4..8......6..2...37...8.7.4.....5 ED=8.9/8.8/8.8`

to dobrichev
i did mine over 2 sets of 400 grids - so i think ......
..... there must be more 23-patterns which don't have many puzzles
..... i am not sure even how to generate a random pattern ....
coloin

Posts: 1733
Joined: 05 May 2005

### Wondering why some patterns have more puzzles

Just musing on why some patterns have significantly more puzzles ....and im sure someone can confirm this one ....

Code: Select all
`diagonal         random+---+---+---+    +---+---+---+|1..|4..|7..|    |12.|...|..8||.2.|.5.|.8.|    |.3.|2.5|..9||..3|..6|..9|    |...|4..|.6.|+---+---+---+    +---+---+---+|2..|6..|9..|    |.6.|.2.|...||.5.|.4.|.2.|    |.79|.3.|..4||..4|..9|..1|    |...|1.9|3..|+---+---+---+    +---+---+---+|4..|5..|1..|    |8..|..2|..1||.1.|.7.|.3.|    |..3|5..|2..||..2|..4|..5|    |9..|8..|.5.|+---+---+---+    +---+---+---+ `

if we look at the non clue cells in the puzzle
we can count up the number of potential U4s there could be
In the diagonal puzzle this is easy - there are 18 pairs of boxes - and each pair could have 3 U4s
I cant do it easily by eye - but there seems more potential spaces in the random puzzle for there to have a U4

Other smaller unavoidable set patterns [U6-U12] could be used too

But this has to be a factor in the number of puzzles
[ minimal / non-minimal]
[ minimal >> non-minimal as clue count decreases]

C
coloin

Posts: 1733
Joined: 05 May 2005

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