## grid game

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

### grid game

Here's a variation on the pattern game. The idea is to take a solved grid and see how many different rated puzzles can be made from it. It may be that every grid has puzzles of every rating in it. Alternatively there may be some solving strategies that require the solution grid to have a certain structure. In this case there may be no grids that have puzzles of every rating or there may be some.

Given the nature of the thread, whether the puzzles are symmetric or minimal is of secondary importance. Let's say that symmetric puzzles trump unsymmetric ones and minimal puzzles trump symmetric puzzles. So minimal, symmetric puzzles are top trumps. Also if you get the chance to choose the grid, try to choose one for some special property.

I don't know that much about the structure of solution grids, so I'll kick things off with the SF grid:

Code: Select all
6 3 9 2 4 1 7 8 5
2 8 4 7 6 5 1 9 3
5 1 7 9 8 3 6 2 4
1 2 3 8 5 7 9 4 6
7 9 6 4 3 2 8 5 1
4 5 8 6 1 9 2 3 7
3 4 2 1 7 8 5 6 9
8 6 1 5 9 4 3 7 2
9 7 5 3 2 6 4 1 8

Here's one of the 17s from that grid ER 1.7 to get things started:
Code: Select all
. . . | . 4 . | 7 . .
. 8 . | . . . | . . .
5 1 . | . . . | . 2 .
------+-------+------
. . . | 8 . . | . . 6
7 . . | . . . | . . .
4 . . | . . . | 2 . .
------+-------+------
3 . 2 | . 7 . | . . .
. . . | . . . | . . .
. . . | . . 6 | . 1 8
ab

Posts: 451
Joined: 06 September 2005

Code: Select all
. 3 .|. 4 .|. . 5
. 8 .|7 . 5|. . .
. . .|. . .|6 2 .
-----+-----+-----
1 . .|. . .|. 4 .
. 9 6|. . 2|. . .
. . 8|. 1 .|. 3 7
-----+-----+-----
3 . .|. . .|. . .
. . .|5 . .|. 7 .
9 . 5|. . 6|. . . ER=7.1

tarek

Posts: 2637
Joined: 05 January 2006

Having started the game I ought to participate
Code: Select all
. 3 9 | . 4 . | 7 . .
2 . . | . . 5 | 1 . .
. . . | 9 8 . | . 2 4
------+-------+------
. . 3 | . . . | . 4 .
7 . 6 | . . . | 8 . 1
. 5 . | . . . | 2 . .
------+-------+------
3 4 . | . 7 8 | . . 9
. . 1 | 5 . . | . . 2
. . 5 | . 2 . | . 1 .
ER=3.0

6 3 . | 2 . 1 | . 8 .
. . . | . . . | . . .
. . . | 9 8 3 | . 2 .
------+-------+------
1 . . | 8 5 7 | . . 6
. . . | . . . | . . .
4 . . | 6 1 9 | . . 7
------+-------+------
. 4 . | 1 7 8 | . . .
. . . | . . . | . . .
. 7 . | 3 . 6 | . 1 8
ER=3.2
ab

Posts: 451
Joined: 06 September 2005

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

SE 1.2 full sym.

. . .|. . .|7 . .
. . .|. 6 5|. . 3
. . .|9 8 3|. . 4
-----+-----+-----
. . 3|. . .|9 . .
. . 6|4 . .|8 5 .
. 5 8|. 1 .|2 . .
-----+-----+-----
. . .|1 . 8|. . .
8 . .|. . .|. . .
. 7 5|. . .|. . .

SE 2.3  no sym.
wintder

Posts: 297
Joined: 24 April 2007

They are \-symmetric but non-minimal. I don't have an ER/SE rating for them, but my solve ran into a range of difficulty. Use any you like and discard the remainder. Some cells occur frequently as clues. It's probably a byproduct of how I generate the puzzles.

Code: Select all
..........84.....3.179....4..3.57......4..85....6.9.37....7.5.9....94....75..64..
.....17...84...1.3.17.....4...857......4.....4..6.9..734....569......3...75..64.8
...2..7...8.76......79836..123...9...96.3.8....8..9...3.217.56.......37.........8
...24.7...847..1...1..836..12.8.....7.6.3......8......342...56.......372.......18
..9.4.7...8.76....5.7.83..4.2..5..4679643..5...8..9...3...........59..7...53.....
.3......5284..51...17....2....85..4....432..1.5..19....4.........15...729...2..18
6.......5.84...19..1.9.......385.94....43...........3..4.1..5...6.5.4..29......18
6....17...84.6.....179.3.2...3..79...9..3.8..4.86..2.73..178.....1....7......6..8
6..241.85.8...5.....7.8..241..8...4.7.6..2...45..19.........5..8.15...7.9.5.....8
63.2..7.52.....19...79....41.3...9.6........1.....9...34.1..56..6....3..9.532....
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

List of minimal puzzles with the rating :
Code: Select all
9.0 # 6.9...7..2......9..1......4..38....6.9.4....1..8.1.....4..785.......43.2.....6.1.
8.9 # .3......5.8..6.1.3..7...6....3.579....64....1.....92....21...6.....943...7.32....
8.6 # 63.2..........5.9...7.8...4......94......2.5..5.6....73.2.78..98..5.43...7.......
8.5 # .3.2..7..2...65.....7.8.62...3...9.6.9.....5.4...1......2.....98.....3....53.....
8.4 # .3.2...8...4.6.1..5.....62.....5.......4..8.......9.3734.1.8...8...9.....75.2..1.
8.3 # 6.....78....76...3.1.......1.3.......96...8.......92.7..2..8..9...5.4.7...5.2.4..
7.9 # ..92..7.........9.5...83..41.3...9.........51.5.6..2...42.78...8....4...9......1.
7.8 # 63................5..983.......57.4....4..8....8...2....21....9.6...4.7.9..3.6..8
7.7 # ....4...5....6...3.1.9....4.2.8.....7.6......45..19..7.4..7..698..5....2..5...4..
7.6 # 63..4..8.......1.3..79......23857.4.7....28.........3..4..7.....615.4..29........
7.5 # 63.......2..7..19..1.98..2..........7964.2...4....9..73.2..8..9.6.5....2......41.
7.3 # .39..1.8.....6....5.7......1.3..79.........5.....1.2.....1.8....6..9...2.7.3..4.8
7.2 # ....417....4....9.5...8.....2...7.4..96...8.....61...........6..6.5...7...5.2...8
7.1 # ......7...8...5..3...9.36.......79....6......4.8.1....3.217..6.......3..9.5.2.41.
7.0 # 6.9....8...47.........83.2...3.......9...28..458..9..7.....8..9.6...........2.41.
6.9 # ..92.1...2..76..9......36.41..8.........3.8....8..92...4..7...9..1.......753.6..8
6.8 # .........2.4...1..5..9.3.2....85.9..7..4..8......1.2...4....5.98..5......7.3....8
6.7 # .3.2.............3...9..6.4.2....94..96.3....45.61.........8....61...37.9...2..1.
6.6 # 639.4...52.......3....8......38....6.....2....5.61......2.7.........43.2.7....41.
6.2 # 63....78.......1..5..9.....1.3..7.46.9......1...........21.8.6..61.943.....3..4..
5.7 # ........5.......9.5.7.83.....3......79.4...514..6....7.421.....8....437......6...
5.6 # .3..4..852.....1..........4...85....7..............2373...7....8.1........5.264..
4.6 # ..9.4...5...76....5....3.2.1.3857...7...3..........2...4.....6.......372..53..4..
4.5 # ..9...7...8..6...3.1.....2..238.7..6...4....14.......7.....8.6......4...9.53..4..
4.4 # ....41..5.8..6..9..1...3.....3.5..46..6..2.51..............8.....1...372.75...4..
4.2 # 6...41.......6...3.......2....8...46.9...2....5..19...3.2.7..6.8......7...5.....8
4.0 # 63............5....179.......3..79.......2.5....61..3...........61..4..2.75.2..18
3.6 # ..9....85..........17..3.2..2...7.467.64...514...19....42....6....59.3.......6...
3.4 # ..9...7......65..3.1..8......3..79.6.......5.4.8......3..1....98...9...2.7.3..4..
3.2 # 6....1.8.2.4...1.....9.3.....3..........3.85..5.6.9......1...698.1.....29....64..
3.0 # ..92.......4..51.......3..41...5....7...3...1..8..92..3..1...69..........7....418
2.8 # ....4.7.......5.9..1.9..62...........9...2.514.861.......17....8....4.7...53.....
2.6 # ..9.41.......6....5.7....2...3.5..4.......85......9..7.4...8..9.615....2.7......8
2.5 # 6..2..78...4.65.....7..3...12.8......9..3.....5...92.....1.8..9.........9.....4..
2.3 # .3...1.....47.5.9.5..9..6..1........7...3.8....8..9.3.....7......1......9...2.4.8
2.0 # .....1..5.8.7.....5......2.....5..4.7.....8...5.619.....2.......61....7....32.4..
1.7 # 6..2..7....4......51........23.5.9..7.6...8...5..1.........8........4.72.7...6.1.
1.5 # .......852..7..........36.4..3.5.9....6.3...14.......7.4.....6.8....4.7.9........
1.2 # .3..4.78.......1.3...9......2......6..6..2.......1..3...2...5.9.....4..2.7.3.6..8
JPF
JPF
2017 Supporter

Posts: 3754
Joined: 06 December 2005
Location: Paris, France

Two more:
[edit: delete garbage]
Mike Metcalf
Last edited by m_b_metcalf on Fri Jun 27, 2008 4:15 am, edited 1 time in total.

m_b_metcalf
2017 Supporter

Posts: 8644
Joined: 15 May 2006
Location: Berlin

SuDoku Explainer 1.2.1 ratings for
daj95376's puzzles
minimal and symmetrical --
Code: Select all
..........84.....3.179....4..3.57......4..85....6.9.37....7.5.9....94....75..64..,7.1

non-minimal ( though symmetrically-minimal ), symmetrical --
Code: Select all
6.......5.84...19..1.9.......385.94....43...........3..4.1..5...6.5.4..29......18,2.8
63.2..7.52.....19...79....41.3...9.6........1.....9...34.1..56..6....3..9.532....,2.8
6..241.85.8...5.....7.8..241..8...4.7.6..2...45..19.........5..8.15...7.9.5.....8,4.5
...2..7...8.76......79836..123...9...96.3.8....8..9...3.217.56.......37.........8,6.6
.....17...84...1.3.17.....4...857......4.....4..6.9..734....569......3...75..64.8,7.2
..9.4.7...8.76....5.7.83..4.2..5..4679643..5...8..9...3...........59..7...53.....,7.3
6....17...84.6.....179.3.2...3..79...9..3.8..4.86..2.73..178.....1....7......6..8,7.7

Pat

Posts: 3497
Joined: 18 July 2005

m_b_metcalf wrote:Two more:
Code: Select all
639001080204760000000000000000000000006400000000019030300078500800500000070000400  9.2

000200785000000000010900000000000900706000000058010207342070009000000000905300008  9.3

I'm not sure your ratings are correct : 2.3 and 2.0

JPF
JPF
2017 Supporter

Posts: 3754
Joined: 06 December 2005
Location: Paris, France

JPF wrote:
m_b_metcalf wrote:Two more:

I'm not sure your ratings are correct : 2.3 and 2.0

JPF

Sorry, you're right -- a bookkeeping error.

Regards,

Mike Metcalf

m_b_metcalf
2017 Supporter

Posts: 8644
Joined: 15 May 2006
Location: Berlin

The number of valid puzzles in the SF grid has been estimated here :

That is 2^81/1.64 = 1.47 x 10^24

The number of minimal non equivalent puzzles is of course far less, but still very large.

I guess that whith such a huge amount of puzzles, we could find all the ER ratings from 1.2 to 11.x ?

Here is an updated list of different ratings for minimal puzzles :
Code: Select all
9.0 # .39....8..8.7.5............1.38.....7.64.2.........2........56.8...9...2.7.3.64..
8.9 # ......78.28............36...2..57...7...3..5.4....92.....1..56...1..4.729.....4..
8.8 # 6.9.417...8.7...9........241...5.9.......285.4...19.......7....8.....37.9....64..
8.6 # .3.24..8..8..651....79........8...46.9.4.........1.2...4......9..159.....7....4.8
8.5 # .3.2..7..2...65.....7.8.62...3...9.6.9.....5.4...1......2.....98.....3....53.....
8.4 # ...24.7...8..6.........3..4.23..7.....6.3.8...5...9.3...2....6986..9.....7.....1.
8.3 # .....17....4.6519.....8...4..38..94...6.......5.....37.4....56.8.........7.3...1.
8.2 # 6..2...8....7.5.....79.36.41.......6......85...8...2....2.....9.6.5.4....7..2.4..
8.0 # ..9.41....8.7...9.51..8.6..1...5..4.............6..2.7...1..5.....5.4.7.9.......8
7.9 # .3..4..8..8..651.....9....4..3...94...6..2..1.58.....7..21.8......5.....9...2..1.
7.8 # 6.......5.8.7.5.9..17.8.............7..43.....5...92..3...7856.8......7....3..4..
7.7 # ....4...5....6...3.1.9....4.2.8.....7.6......45..19..7.4..7..698..5....2..5...4..
7.6 # .......85..47.5.9..1...3......8.......6.....1458..9..73..1...69.....4.7.9.53.....
7.5 # ..9..1..5.8.7.5..3......6..1.........9.43..5...8...2.73.2........1.9.........64..
7.4 # 6...4178........9.....83....2..5.....9.....514..6.92....2..8....61.9..........418
7.3 # ....4.7...8.7....3.....36....3.5.9..7.....8.14...1..3....1...6.8..59..7.9.....4..
7.2 # .............6...35.7..3.2....8..9..7..43.8.145.......3.2.785...6..9.3.........1.
7.1 # ..9..1....8.7.5.9......36.4..38.7....9.4...5...8.1.....4.......8.1..43......2..1.
7.0 # 6.9....8...47.........83.2...3.......9...28..458..9..7.....8..9.6...........2.41.
6.9 # ..92.1...2..76..9......36.41..8.........3.8....8..92...4..7...9..1.......753.6..8
6.8 # ..9.....528..6.19......36..1.....9.....4.2.....8..9....4...8....6.5....2.7......8
6.7 # 6....1.85....6.1.....9.36..1.3.....6.9.4....1..8.....7.4.......8.1..43.......6...
6.6 # .3.2.......4.6519..........1.......67..4...51..8...2...4..7.5.....5.43..97...6...
6.5 # ..92..7.....7..1.351..8.6..12..5...6.......5.4...1...73.........6.......9753.....
6.2 # 63....78.......1..5..9.....1.3..7.46.9......1...........21.8.6..61.943.....3..4..
5.9 # ..9.4.7.5....6...351..8...4.23...94...............92...42..8...8....43...7...6...
5.7 # .3..4.7.........9....98...4...8.7..6.9........5.6...3.3......6.86.5.......532..18
5.6 # .....1.8..847.5...5.....6..1..8...4..9.43.....5.....3...2..8..9........2.7...6..8
5.0 # ......7.5...7.5....1.9...2..............3..514.8...2..34.1.8....6.5.4.7.9.....4..
4.6 # ..9.4...5...76....5....3.2.1.3857...7...3..........2...4.....6.......372..53..4..
4.5 # .3..41..........935.7...6......5794.7.....8....86..2..3.........6.59........2.4.8
4.4 # 6.......528.7.......7..3..4.2..5...6.9.4....1..........4...85....1..4.7.9....6.1.
4.2 # 6...41.......6...3.......2....8...46.9...2....5..19...3.2.7..6.8......7...5.....8
4.0 # .392..7....4.6.........36.......7...........1.5.6.9.3...2...5....1.94...9.5..64..
3.8 # .3.2..7.5........3.1.9..62........46..........5.61.2.73...7....8.1..4...9..3...1.
3.6 # ..92..7......6.1..51......4.23.......9....8.1....19..73......6....594.....5.....8
3.4 # ...24...5....6..9......3....2..5......6...8.......9.3734...8...8....4..29......1.
3.2 # 6..............19..1.....24.238.7....9..........6...373....8569...59........2..1.
3.0 # ........52847......1....62.1.....9...9..32..1..86......4...8......5.4.7.9.....41.
2.8 # .3...1...2...65.9......362...3.5.......4....1.5....2...4.1.8..9......37.9........
2.6 # ....417..2......9.....83.....3...9.6.96...8..4...1....3.2...5.........7..7.....18
2.5 # 6..2..78...4.65.....7..3...12.8......9..3.....5...92.....1.8..9.........9.....4..
2.3 # .3...1.....47.5.9.5..9..6..1........7...3.8....8..9.3.....7......1......9...2.4.8
2.0 # .....1..5.8.7.....5......2.....5..4.7.....8...5.619.....2.......61....7....32.4..
1.7 # 6..2..7....4......51........23.5.9..7.6...8...5..1.........8........4.72.7...6.1.
1.5 # .......852..7..........36.4..3.5.9....6.3...14.......7.4.....6.8....4.7.9........
1.2 # .3..4.78.......1.3...9......2......6..6..2.......1..3...2...5.9.....4..2.7.3.6..8

JPF
JPF
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Posts: 3754
Joined: 06 December 2005
Location: Paris, France

JPF wrote:I guess that whith such a huge amount of puzzles, we could find all the ER ratings from 1.2 to 11.x ?

JPF

I guess that there are an infinite number of twin primes

If you find a puzzle for every rating from 1.2 to 11.x then it is proved. Your contribution to this thread has certainly helped in that respect.

Of course if you don't find a puzzle for every rating nothing is proved. Proving that a grid doesn't have a puzzle for a given technique would be very hard. It would be an interesting result though
ab

Posts: 451
Joined: 06 September 2005

ab wrote:I guess that there are an infinite number of twin primes
Maybe...

There is a class of puzzles which is easy to study : the full symmetrical puzzles included in this grid.

There are exactly 16294 fully symmetrical valid puzzles, but none of them is absolutely minimal.
The number of clues varies between 28 and ... 81

The highest ER rating is 7.3 for this 33 clues puzzle :

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

Here are the possible ER ratings :
Code: Select all
7.3 # 63..4..852.4...1.3.17...62....8.7...7...3...1...6.9....42...56.8.1...3.297..2..18
7.2 # 6..2.1..5...........79836..1.3...9.6..6...8..4.8...2.7..21785...........9..3.6..8
7.1 # ..9.4.7...8.....9.5..9.3..4..3.5.9..7..4.2..1..8.1.2..3..1.8..9.6.....7...5.2.4..
6.9 # 63.2.1.852..7.5..3....8....12.....46..6...8..45.....37....7....8..5.4..297.3.6.18
6.6 # 63.2.1.852.......3..7...6..1..857..6...4.2...4..619..7..2...5..8.......297.3.6.18
6.5 # 63.2.1.8528..6..93.........1..8.7..6.9.....5.4..6.9..7.........86..9..7297.3.6.18
5.6 # .3.....8.2..765..3..7...6...2.857.4..9.4.2.5..5.619.3...2...5..8..594..2.7.....1.
4.5 # ..92.17......6....5.7...6.41..8.7..6.9..3..5.4..6.9..73.2...5.9....9......53.64..
4.4 # 6.......5..47.51...1.....2..2.857.4....4.2....5.619.3..4.....6...15.43..9.......8
4.2 # ..9.4.7...84...19.51.....24...8.7...7...3...1...6.9...34.....69.61...37...5.2.4..
3.8 # 6..2.1..5.8..6..9...79.36..1.3...9.6.9.....5.4.8...2.7..21.85...6..9..7.9..3.6..8
3.2 # 6..2.1..5.8.....9...79.36..1.3...9.6....3....4.8...2.7..21.85...6.....7.9..3.6..8
3.0 # ...2.1....8..6..9...79836..1.3...9.6.96...85.4.8...2.7..21785...6..9..7....3.6...
2.8 # .3..4..8.2.......3..79.36....38579..7..4.2..1..86192....21.85..8.......2.7..2..1.
2.6 # 6..2.1..5..4...1...17...62.1..8.7..6....3....4..6.9..7.42...56...1...3..9..3.6..8
2.5 # 6.92.17.5.........5.7...6.41..8.7..6....3....4..6.9..73.2...5.9.........9.53.64.8
2.3 # ..92.17...8..6..9.5.......41...5...6.9.432.5.4...1...73.......9.6..9..7...53.64..
2.0 # 6..241..5.8.....9....9.3...1.3...9.67.......14.8...2.7...1.8....6.....7.9..326..8
1.7 # 6...4...5..4.6.1...17...62....8.7...79.....51...6.9....42...56...1.9.3..9...2...8
1.5 # ..92417...........5.7...6.41..8.7..67.......14..6.9..73.2...5.9...........53264..
1.2 # ...241....8.....9...79.36..1.3...9.67.......14.8...2.7..21.85...6.....7....326...
1.0 # 6392.17852.47.51.35179.36241238.7946.........4586.92373421.85698.15.43.29753.6418
0.0 # 639241785284765193517983624123857946796432851458619237342178569861594372975326418

JPF
JPF
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Posts: 3754
Joined: 06 December 2005
Location: Paris, France

More :

Minimal puzzles

diagonal and antidiagonal symmetry, 27 clues
Code: Select all
. 3 . | 2 . . | . 8 5
2 8 . | . . . | 1 . 3
. . 7 | . . . | . 2 .
-------+-------+-------
1 . . | 8 . 7 | . . .
. . . | . 3 . | . . .
. . . | 6 . 9 | . . 7
-------+-------+-------
. 4 . | . . . | 5 . .
8 . 1 | . . . | . 7 2
9 7 . | . . 6 | . 1 .     ER=8.9

horizontal and vertical symmetry, 28 clues
Code: Select all
. 3 9 | . 4 . | 7 8 .
2 . . | . . . | . . 3
5 . 7 | . . . | 6 . 4
-------+-------+-------
. . . | 8 5 7 | . . .
. . . | . . . | . . .
. . . | 6 1 9 | . . .
-------+-------+-------
3 . 2 | . . . | 5 . 9
8 . . | . . . | . . 2
. 7 5 | . 2 . | 4 1 .     ER=8.9

Symmetrically-minimal

diagonal and antidiagonal symmetry, 28 clues
Code: Select all
. 3 . | 2 . . | . . 5
2 . 4 | . . 5 | . . .
. 1 . | . 8 . | . . .
-------+-------+-------
1 . . | . 5 7 | . 4 .
. . 6 | 4 . 2 | 8 . .
. 5 . | 6 1 . | . . 7
-------+-------+-------
. . . | . 7 . | . 6 .
. . . | 5 . . | 3 . 2
9 . . | . . 6 | . 1 .      ER=9.0

JPF
JPF
2017 Supporter

Posts: 3754
Joined: 06 December 2005
Location: Paris, France

Interesting thread....so there are no fully symmetric minimal puzzles in the SF grid.

This has implications for the total number of fully symmetric puzzles . Perhaps also explains why mauricio never found the elusive 36 clue fully symmetric minimal puzzle !

Each grid has around 10^14 - 10^15 minimal puzzles according to estimates - which is in line with JPFs estimates of total minimal and non-minimal puzzles.

In an effort to explore the extent of the distribution of "hard puzzles".

Many of our hardest puzzles have 21 clues.

I estimate, that on average, there are about 50 million 21-puzzles per grid. [estimated by the frequency of duplicate generation]

The SF grid probably has significantly more than this.

From the archives this is a distibution of the clue counts of puzzles generated by dukuso

Code: Select all
dukoso wrote:

here is some statistics, starting from a full grid and generating 1e6
random locally minimal sudokus from it.

1) one grid from each G-class at random
2) Gordon's grid with 29 17s, the SF grid
3) our canonical grid,(1,1,1-1,1,1)

Code:

clues , 1)     2)      3)
----------------------------
17,     0       0       0
18,     0       0       0
19,     0     4.3       0
20,    59     182       0
21,  2428    6051      85
22, 33966   61826    1775
23,170727  227480   21648
24,342620  352289  116766
25,298349  248568  286836
26,122691   86061  329853
27, 25237   15908  185028
28,  2733    1547   50469
29,   205      74    7040
30,   7.6     8.6     486
31,     0       0      12
32,     0       0     2.4
-------------------------------
aver.24.38   24.10   25.72

The method of puzzle production skews the distribution away from large puzzles.

From this The SF appears to have approx x2 more 21-puzzles than average.

At the other end of the scale there are numerous 34-puzzles per grid to be found - if you try hard enough. In the random grid I studied - I gave up when it became obvious there was an increasingly expanding amount of puzzles - and this was only in one region in the grid. Interestingly many of the 34-puzzles have a highish ER.

Code: Select all
+---+---+---+
|6.9|.4.|...|
|.8.|.65|..3|
|5.7|983|6..|
+---+---+---+
|...|...|..6|
|79.|.3.|85.|
|...|..9|23.|
+---+---+---+
|.4.|..8|...|
|8..|.94|37.|
|..5|326|4..|
+---+---+---+  34 clue minimal in SF grid

My [hardly earth-shattering] conclusion [based on the estimate of 5*10^7 * 5*10^9 ~ 2*10^17 for the total number of 21-puzzles] was that it was unlikely that we have fully explored all the possible 21-puzzles !

C
coloin

Posts: 1666
Joined: 05 May 2005

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