The New Sudoku Players' Forum

[coloin]

This puzzle.....

Code: Select all

+---+---+---+
|...|5..|1..|
|.98|...|...|
|...|...|.4.|
+---+---+---+
|5..|123|.7.|
|...|456|...|
|...|789|...|
+---+---+---+
|...|...|2.9|
|1..|3..|...|
|...|...|8..|
+---+---+---+

has 9 clues in box 5 and a valid puzzle is produced by adding just 12 more clues.

A little thought will show that each of these puzzles has 6^6 * 2 isomorphic equivalents with the minlex box5.

Thus the search space could be less than it would otherwise appear.

Probably very few essentially different < 100 ? of these puzzles

I am not sure which is the best way to canonicalize these puzzles for reference purposes. ?

Also , potentially there could be a "9plus13" or "9plus14" puzzle which is superhard.

Anyhow, whilst the patterns game is regrouping - perhaps you can find one ?

C

[coloin]

Puzzles found

Code: Select all

01# 000000001000000002003004000000005060010000000270000300000189000000247000009563400 # gfroyle
02# 000000000000000012003045000000001000000206000007000800128000000394000507675000000 # gfroyle
03# 000000001000000020003004000000005300010000600270000000000126000000489000000753460 # gfroyle
04# 000000001000000020000034000000500006002007000004000300231000040478000000695100000 # gfroyle
05# 000000001000000023000004000000020500000130000006700400127000000684009000935000000 # coloin

C

[ronk]

This puzzle.....

Code: Select all

+---+---+---+
|...|5..|1..|
|.98|...|...|
|...|...|.4.|
+---+---+---+
|5..|123|.7.|
|...|456|...|
|...|789|...|
+---+---+---+
|...|...|2.9|
|1..|3..|...|
|...|...|8..|
+---+---+---+

has 9 clues in box 5 and a valid puzzle is produced by adding just 12 more clues.

A little thought will show that each of these puzzles has 6^6 * 2 isomorphic equivalents with the minlex box5.

Bands 1 & 3 and stacks 1 & 3 can be swapped too, so wouldn't that be 6^6 * 2^3 ?

[m_b_metcalf]

Also , potentially there could be a "9plus13" or "9plus14" puzzle which is superhard.

Your puzzle is SE=6.7. The best I could do with a quick run was some "9plus14"s, of which the hardest was:

Code: Select all

 1 . . . . . 2 3 .
 . 4 . 5 . 6 . . .
 . . . . . . . . .
 . . . 4 7 5 . . 8
 . . . 6 2 9 . . .
 2 . . 3 1 8 . . .
 . 6 4 . . . . . .
 . 8 . . . . 7 1 .
 . . . . . . 9 . .   ED=8.9/1.2/1.2

Hardly 'superhard', I'm afraid.

Regards,

Mike Metcalf

[ronk]

A 9plus13 "diamond"...

Code: Select all

 . . . 8 . . 3 . .
 . 9 5 . . . . . .
 . 7 . . . . . 4 .
 6 . . 1 2 3 . 7 .
 . . . 4 5 6 . . .
 . . . 7 8 9 . . .
 . . . . . . 2 . 9
 1 . . 3 . . . . .
 . . . . . . 5 . .    ED=6.7/6.7/6.7

[coloin]

Code: Select all

000000001000000002003004000000005060010000000270000300000189000000247000009563400
000000000000000012003045000000001000000206000007000800128000000394000507675000000
000000001000000020003004000000005300010000600270000000000126000000489000000753460
000000001000000020000034000000500006002007000004000300231000040478000000695100000

Here are the minlex morphs of 3 more "9plus12" puzzles [including the first example]

Ive attributed them to gfroyle as they are derived from 4 of the 17-puzzles with 5 clues in a box

but there are more to be found

indeed the isomorphs do have a furthur 2^2 ways ........ well pointed out.

so we probably should really be able to find them all then - given that there are so many isomorphs of the same puzzle with the same box5.

I might have to reconsider the possibility that there will be a superhard puzzle with a full box 5. Bivalues are difficult to prevent.

Although mauricio gave us this

Code: Select all

+---+---+---+
|6..|..2|.59|
|52.|.4.|.1.|
|..3|5..|2..|
+---+---+---+
|3..|194|5..|
|.1.|658|.3.|
|..5|273|..1|
+---+---+---+
|..4|..5|1..|
|.3.|.2.|.45|
|75.|4..|..8|
+---+---+---+  Mauricio's non-minimal puzzle SE 11.0

C

[m_b_metcalf]

I might have to reconsider the possibility that there will be a superhard puzzle with a full box 5. Bivalues are difficult to prevent.

Although mauricio gave us this

Code: Select all

+---+---+---+
|6..|..2|.59|
|52.|.4.|.1.|
|..3|5..|2..|
+---+---+---+
|3..|194|5..|
|.1.|658|.3.|
|..5|273|..1|
+---+---+---+
|..4|..5|1..|
|.3.|.2.|.45|
|75.|4..|..8|
+---+---+---+  Mauricio's non-minimal puzzle SE 11.0

C

Funny, if I take away a random set of redundant clues, meaning the puzzle is no longer a diamond (pearl?), I get

Code: Select all

 6 . . . . . . 5 9
 5 2 . . 4 . . 1 .
 . . 3 . . . 2 . .
 . . . 1 9 4 5 . .
 . 1 . 6 5 8 . 3 .
 . . 5 2 7 3 . .
 . . 4 . . 5 1 . .
 . 3 . . 2 . . 4
 7 . . . . . . . 8   ED=11.2/1.2/1.2, 30 clues

Maybe the 11.0 came from an old version of SE.

Regards,

Mike Metcalf

[m_b_metcalf]

Here are some (not very exciting) "9plus13"s, based on over 3000 '4 in a box' puzzles in Gordon's file, for the record:

Code: Select all

000100230045000000000000000000654000300217600000893000260700000800000090000000004  ED=1.5/1.5/1.5   
100000203040500000000000000060789040000625800000413000000800760003000000500000000  ED=2.0/1.5/1.5   
100020030000040050600000000700652800000137000050894000093000000000000700000000600  ED=2.6/1.5/1.5   
010203000000000045000000000000467000400352080020819300000100200500000000008000000  ED=3.0/1.2/1.2   
000100020034000000000000000000564000200718090000239000500800600100000004000002300  ED=4.6/1.2/1.2   
000010023405000000000000000000586100000473000030291000600000500020030000000700080  SE=7.2/1.2/1.2   

Regards,

Mike Metcalf

[coloin]

Just to show that a "9plus12" is possible without having a 17 in it.
Here is a sparse example

Code: Select all

+---+---+---+
|..7|3..|1..|
|8..|...|4..|
|5.9|...|...|
+---+---+---+
|.6.|123|...|
|...|456|...|
|...|789|...|
+---+---+---+
|.1.|...|...|
|...|...|.9.|
|...|...|.58|
+---+---+---+  #05

And yes mauricio's puzzle is 11.2

C

[coloin]

Actually the previous puzzle shows that 8 clues can be added to make a valid puzzle to two complete boxes [B1and B5].

Code: Select all

+---+---+---+
|647|3..|1..|
|831|...|4..|
|529|...|...|
+---+---+---+
|.6.|123|...|
|...|456|...|
|...|789|...|
+---+---+---+
|.1.|...|...|
|...|...|.9.|
|...|...|.58|
+---+---+---+

We do know that 3 clues can be added to make a valid puzzle with 3 complete boxes [B1,B5,B9]

Here are 55 more 9plus12s.

Code: Select all

...........8.9..3.......16....123......456.1...2789.........2.916.......4........
..........69.........5..14.58.123......456......789.........2.64..3...........8..
..........98.........5....44..123....3.456......7892...........1..6....7..2...8..
.........8.2.........3..14....123......456...6..789..5.1.5.............6.......98
.......2.8...........3..14..4.123......456...6..789....1.5.............6.......98
......1........4.5..6..8...75.123......456......789....1.5............6........98
......1.....6..7..2.9.........123......456..2...789...41.5.............96.......8
......1.2.5........89.........1234...9.456......789..........9........561..2.....
......1.3.5........8..6....7..1236.....4563.....789...3............4..5.1........
......12..5.....3..8.9.....7..123.4....456......789...2............3.....6......8
......12.8........5.6.........123.7....456...6..789....1.2.............6......9.8
......2...5...8..3.8.......7..1236.....456......789..........98..7....5.2........
......21..5.....3..8.9.....7..123.4....456......789............2...3..........6.8
......21..5.....3..8.9.....7..123.4....456......789...2............3.....6......8
......27..5.....3..8.9.....7..123.4....456......789............2...3.....6......8
......27..5.....3..8.9.....7..123.4....456......789...2............3.....6......8
......27..5.....3..8.9.....7..123.4....456......789...3............3.....6......8
......27..5.....3..8.9.....7..123.6....456......789...2............3.....6......8
......41..5.....3..8.9........123.4....456......789...3........4..23............8
......41..5.....3..8.9.....7..123.4....456......789............4...3......6.....8
......41..5.....3..8.9.....7..123.4....456......789...4............3......6.....8
.....21............54......7..1236.....456......789...3....1..........582......9.
.....41..8.......5..9.........123.4....456...5..789....1........47....9.........8
...3.....5........68......5...1237.....456......789.......6..98..7........3....6.
...3....2.5........9.......7..1234.....456......789...3..2...........65.1......9.
...3..1....8.....92.6.........1234.....456......789...71..............9........86
...3..1...5.....8..........7..1234.....456......789.....7....9........583.1......
...3..1...54.......9......57..123......456......789..........9........561..2.....
...3..1..25...8....8.......7..1234.....456......789..........98.......5.1........
...3..1..5.8........9.........123......456.8....789....1.2......4.....9........68
...3..1..8.9...3..5.........4.123......456......789....1..............9.....9..54
...3..1..8.9...3..5.........4.123......456......789....1..............94....9..5.
...3..1.4..........85......7..1234.....456......789...................561..2...9.
...3..1.4.5........8.......7..1234.....456......789..........9........561..2.....
...3..12.8...............4..4.123......456...6..789....1.5.............6.......98
...3..15.8........7.....4.....123......456......789..2..........1....5...5..7...8
...3..17.......5..8...........123......456...6..789..2.1.5.............6.9......8
...3..17..5........8.......7..1234.....456......789..........98........64..2.....
...3..2............56....9.7..1235.....456......789...3.......8.......6.1..2.....
...3..2....5.......89.........1234.....456......789...3............9.65.7......9.
...3..2....5.......89.........1236.....456....4.789...3............9..5.7.......9
...3..2...5..............8.7..1234.....456......789....8.....9.......65.1..8.....
...3..2...5........89.........1237.....456......789...3......9.....9.65.7........
...3..7...5........82..5...7..1234.....456......789...........8.......564........
...3..7.1.5.6......8.......7..1234.....456......789..........98.......5.4........
...6..1........7..2.9.........123......456..2...789...47.5.............96.......8
...6..1....8......2.9......7..1234.....456......789....1.5............9........68
...6..1...52..8....8.......7..1234.....456......789..........98.......5.1........
...9..1........4..8.6.........123...9..456...6..789..2.4.5.............6........8
..7.9..3................16....123......456.1...2789.........2.916.......4........
.2.....4..9..........5..17....123...3..456......789.........2..4..6...........8.9
.26.......9....8.....5...4....123......456.7....789.........2.91.4..............8
.52..8......6..1...8.......7..1236.....456......789..........98.......5.1........
.6.........9.........5..14.58.123......456......789.........2.64..3...........8..
62........9....8.....5...1....123......456.7....789.........2.91.4..............8

Fairly unspectacular puzzles, derived from a [surprisingly] still expanding collection of 9plus13s [400,000 of them]

C

[coloin]

Despite the number of isomorphic ways to represent a puzzle with a full central box - I am still surprized at the increasing number of different puzzles there are.... [9plus13, 9plus14.......... etc]

However it probably stops around the 9plus25 mark, a quick look i found this one.

Code: Select all

+---+---+---+
|...|...|3..|
|6..|5.8|.4.|
|..7|.34|.6.|
+---+---+---+
|...|123|.9.|
|1.9|456|..3|
|3.5|789|...|
+---+---+---+
|9.6|8.7|...|
|7.1|..5|.86|
|...|...|..9|
+---+---+---+

those extra 25 clues are all essential.
and that perhaps induces a 9plus26 game - if anyone were interested.......

given that every true minimal puzzle can have 9 ways to fill a box
and several ways to remove unnessesary clues

thats a lot of puzzles..... but i suppose i new that already.

edit - on return from pub.........
well a rough guess of the number of different puzzles would be
9 * 5e9 * 30^8 ~ 3e22

30 might be the average number of ways you can remove clues from a single box

C

[m_b_metcalf]

and that perhaps induces a 9plus26 game - if anyone were interested.......

Is this what you meant?

Code: Select all

 . . . . . . . . .
 . 4 9 8 7 6 . . .
 . . . 9 . . 8 . .
 . 6 . 5 9 2 7 . .
 . 5 8 7 6 1 . 2 4
 . 2 7 4 3 8 5 . 6
 . . 5 2 . 9 . . 7
 . 9 2 . . . . 8 .
 . . . 1 8 . . . .
  No. fixed: 35

And one from your own grid:

Code: Select all

 . . . . . . . . .
 . 1 . 5 7 8 9 . .
 . . . 9 . 4 1 . 5
 . . 8 1 2 3 5 9 .
 . . 9 4 5 6 8 . .
 . 2 . 7 8 9 . 1 4
 . 5 . . 4 . . . .
 . 3 1 2 9 5 . . .
 . . 2 3 . . . . .

Regards,

Mike Metcalf

[m_b_metcalf]

... and, for my next trick:

Code: Select all

 . . . . . . . . .
 . . 3 5 7 8 . . 2
 . 8 . 9 3 . . 6 5
 . 6 8 1 2 3 5 . 7
 . 7 . 4 5 6 8 . 3
 . 2 . 7 8 9 6 . .
 . 5 6 8 4 . . . .
 . . 1 . 9 5 . . .
 . . . 3 . . . . .                  ED=7.3/1.2/1.2
  No. fixed: 36 (= 9plus27)

Your grid, again.

Regards,

Mike Metcalf

[m_b_metcalf]

They're coming thick and fast:

Code: Select all

 . . . . . . . . .
 . . 2 . 5 . 4 . 8
 . 6 . 4 . 2 1 . .
 . 9 . 2 1 7 . 8 .
 . 8 7 3 9 4 5 . .
 . 3 4 8 6 5 2 . 9
 . . 9 7 . 3 . . .
 . 4 . . 2 9 . . .
 . . . 5 4 1 . . .  9plus26, random

 . . . . . . . . .
 . . 5 9 . 4 2 7 .
 . . . 5 8 2 . . 9
 . 8 . 4 2 9 . 1 .
 . 5 1 3 7 8 . 2 .
 . 9 2 6 5 1 . 8 7
 . . 4 . 9 6 . . .
 . 7 . 2 . 5 . . .
 . 1 9 . . . . . .  9plus26, random

 . . . . . . . . .
 . 7 1 4 2 8 . . .
 . . . . 7 6 4 1 .
 . 6 4 3 9 2 7 5 .
 . 3 . 7 8 1 6 4 .
 . 1 . 6 4 5 3 9 .
 . . 7 2 6 4 . . .
 . . . 8 . . 9 6 .
 . . . . . . . . .  9plus26, random

 . . . . . . . . .
 . . 3 5 . 8 . 4 .
 . 8 7 9 3 . . . .
 . 6 8 1 2 3 . 9 .
 . . 9 4 5 6 8 2 .
 . 2 . 7 8 9 6 . .
 . 5 6 . 4 7 2 3 .
 . 3 . . 9 5 4 8 .
 . . . . . . . . .  9plus27, coloin's grid

 . . . . . . . . .
 . 1 . 5 7 8 9 4 .
 . 8 . 9 . 4 1 . .
 . . . 1 2 3 5 9 .
 . . . 4 5 6 8 . .
 . 2 5 7 8 9 6 1 .
 . 5 . 8 4 7 . . .
 . 3 1 2 9 . 4 8 .
 . . . . . . . . .  9plus27, coloin's grid

 . . . . . . . . .
 . . . 5 7 8 . 4 .
 . 8 . . 3 4 1 . .
 . 6 8 1 2 3 5 9 .
 . . 9 4 5 6 8 2 .
 . . 5 7 8 9 6 1 .
 . 5 6 8 . 7 2 . .
 . 3 1 2 9 . . . .
 . . . . . . . . .  9plus27, coloin's grid


Regards,

Mike Metcalf

P.S. Edited a few hours later because I can't count properly.

[coloin]

Thanks for that.I cheated of course, because i took a 39 puzzle, filled up a suitable box and deleted clues to minimality.

And extended to a 9plus28.

Code: Select all

+---+---+---+
|..2|...|...|
|8..|..2|...|
|95.|.6.|...|
+---+---+---+
|4..|123|.9.|
|1..|456|7..|
|.25|789|41.|
+---+---+---+
|2.1|698|54.|
|.4.|.1.|...|
|.98|2.4|1..|
+---+---+---+

in any one grid there are very many puzzles.

it would appear that 9plus12s are rarer than minmal 17 puzzles.

C