The New Sudoku Players' Forum

[coloin]

Well it seems unlikely !.....

...............................
11.1 Dynamic Cell Forcing Chains(+Multiple Forcing Chains)
11.2 Dynamic Double Forcing Chains(+Multiple Forcing Chains)
11.3 Dynamic Contration Forcing Chains(+Dynamic Forcing Chains)
11.4 Dynamic Region Forcing Chains(+Dynamic Forcing Chains)
11.6 Dynamic Cell Forcing Chains(+Dynamic Forcing Chains)
11.7 Dynamic Double Forcing Chains(+Dynamic Forcing Chains)

Eioru seem to know whats next

But it seems to end at 11.7................

It is interesting that our pattern was discussed very early on in the hardest thread pages 13 and 14 !

Code: Select all

+---+---+---+
|X..|.X.|..X|
|.X.|...|.X.|
|..X|...|X..|
+---+---+---+
|...|X.X|...|
|X..|.X.|..X|
|...|X.X|...|
+---+---+---+
|..X|...|X..|
|.X.|...|.X.|
|X..|.X.|..X|
+---+---+---+

here

Useful observation: Puzzles in the X-pattern that satisfy all of the following conditions - 1. All digits in upper band differ (= first seven nonzero digits). 2. All digits in lower band differ (= last seven digits). 3. All digits in left stack differ (= the three left columns). 4. All digits in right stack differ (= the three right columns). 5. Suexrating > 400. - More than 90% of such puzzles were rated 9.0 or above by Explainer (of the about 200 checked). Very effective screening criteria for finding hard puzzles, once it was established.

C

[JPF]

Some new puzzles :

11.3 # 100000002003400050060000700000030040000806000009540000020000100700000006005090080 # JPF 04/10
11.1 # 100000002030400050006000700080300000000015000000809040700000600050008090002000001 # JPF 04/10
11.1 # 100000002030400050006000700080300000000069000000805040700000100050008090002000006 # JPF 04/10
11.1 # 100000002030400050006000700050300000000018000000509040700000600090003080002000001 # JPF 04/10
11.0 # 100000002003400050060000700000050040000301000008940000020000100700000006005090030 # JPF 04/10
11.0 # 100000002030040050006000700000438000040560090000900000002000100050080030700000006 # JPF 04/10
11.0 # 100000002030400050006000700080300000000019000000805090700000600040008030002000001 # JPF 04/10
11.0 # 100000002003400050060000700000080040000301000008540000020000600700000001005090030 # JPF 04/10
10.6 # 100000002030040050006000700000354000040860090000900000002000100050080030700000006 # JPF 04/10
10.5 # 001020000300004000050600070080900005002003000400010000070000038000800069000000200 # JPF 04/10

List here.

Code: Select all

11.2 # 100000002090400050006000700080903000000070000000850030700000600050009080002000001 # JPF 04/07 
11.2 # 1.......6.2.5...4...3...7...4.8.5.......3........24.8.7.....3...5...9.2...6.....1 # isomorph ?

Yes.

JPF

[tarek]

..it also requires more than the usual amount of propositions to solve
if you let it crank a few minutes it should rate at 100008
so much for the intended 99999 limit

Thanx

ER=11.3
Here are their canonical representations :
.
.
It was on my waiting list for ER evaluation.

I hope I didn't spoil any of the fun, I'll promise to PM next time
From experience....The waiting list issue is just a big headache, it just ends up getting bigger & bigger.... I ended up deleting most of the stuff I had

SE rating is tooooooooo slow , I will therefore rely on gsfr & suexrat9 & revisit with ER.

tarek

[tarek]

These are some of my findings (as they rely on similar strategies to what coloin proposed, please let me know if any isomorphs appear)

The list will be revisited with ER (whenever that happens)...the candidates were shortlisted as having (gsfr>=9800 AND sxr>=1000 AND M2 AND minimal):

PUZZLE________________________________________________________________________________________ #_________ Q2_ QHDST_ SXR9 ER
100000009030004070006000200050940000000017000000305080002000600040700030900000001 ULTRA0000 106 100002 1051 105
300000007050008060002000900010460000000020000000107040009000200080600050700000003 ULTRA0001 083 099995 1130 105
100000007020004030005000800090603000000400000000908060008000500040300020700000001 ULTRA0002 089 099994 1451 108
600000002080001070003000400090082000000067000000900050004000300010700080200000006 ULTRA0003 083 099994 1028 109
800000005010003070006000400020401000000009000000307090004000600030700010500000008 ULTRA0004 110 099993 1502 107
300000007020004060005000800090802000000001000000906010008000500040600020700000003 ULTRA0005 100 099992 1446 108
500000004080006090001000200070308000000050000000790030002000100060900080400000005 ULTRA0006 086 099992 1403 107
900000005040001030002000700060083000000504000000016080007000200010300040500000009 ULTRA0007 102 099992 1268 112
200000006050008090007000300040015000000609000000084010003000700080900050600000002 ULTRA0008 095 099992 1260 107
600000002080005090003000400070410000000860000000057010004000300050900080200000006 ULTRA0009 089 099992 1240 107
300000001020006090005000800070204000000867000000009040008000500060900020100000003 ULTRA0010 126 099992 1159 107
400000003080002060007000900010508000000701000000026050009000700020600080300000004 ULTRA0011 106 099992 1133 107
700000005040001030002000900060008000000946000000103080009000200010300040500000007 ULTRA0012 110 099992 1126 106
200000006090007040001000300050804000000205000000079080003000100070400090600000002 ULTRA0013 106 099992 1126 106
400000008050002090001000600070503000000060000000790030006000100020900050800000004 ULTRA0014 082 099991 1494 106
400000001090006020008000500070008000000319000000760030005000800060200090100000004 ULTRA0015 097 099991 1045 107
600000002080005070001000900040150000000027000000408030009000100050700080200000006 ULTRA0016 104 099991 1016 107
800000001050009040003000600070056000000980000000704020006000300090400050100000008 ULTRA0017 086 099991 1005 105
400000003010002070008000500060209000000100000000680090005000800020700010300000004 ULTRA0018 083 099876 1166 105
600000002070008050009000100040003000000147000000805030001000900080500070200000006 ULTRA0019 116 099865 1199 106
900000004060007050002000300080360000000791000000085010003000200070500060400000009 ULTRA0020 007 099863 1022 104
500000008030004070009000100020607000000083000000042060001000900040700030800000005 ULTRA0021 098 099860 1002 107
600000002080005010003000400070928000000500000000016090004000300050100080200000006 ULTRA0022 103 099856 1257 105
300000009080001070002000600050840000000503000000107040006000200010700080900000003 ULTRA0023 211 099854 1219 106
100000004020006090005000800030650000000372000000098070008000500060900020400000001 ULTRA0024 099 099854 1072 106
700000003060009040008000500010602000000150000000940020005000800090400060300000007 ULTRA0025 129 099854 1024 104
900000001030004070006000200050460000000853000000072080002000600040700030100000009 ULTRA0026 100 099853 1123 106
400000003070002060008000500010406000000280000000170090005000800020600070300000004 ULTRA0027 009 099853 1052 105
500000008030007040001000900020603000000725000000800060009000100070400030800000005 ULTRA0028 115 099845 1111 106
400000009070008030006000100050702000000065000000003020001000600080300070900000004 ULTRA0029 078 099845 1026 106
100000009020004030005000800070400000000630000000725060008000500040300020900000001 ULTRA0030 109 099843 1155 106
200000006010005080007000300090004000000170000000593040003000700050800010600000002 ULTRA0031 103 099842 1055 105
300000007020006090005000800010420000000103000000609040008000500060900020700000003 ULTRA0032 104 099840 1122 106
400000003090006070008000500010900000000210000000674020005000800060700090300000004 ULTRA0033 011 099836 1180 104
400000003060007090008000500020709000000610000000280010005000800070900060300000004 ULTRA0034 110 099820 1175 104
600000002070008050001000900040203000000786000000005030009000100080500070200000006 ULTRA0035 099 099810 1121 105
300000005020007040001000900080036000000028000000704060009000100070400020500000003 ULTRA0036 103 099802 1184 106
900000005040008030002000100060980000000647000000050070001000200080300040500000009 ULTRA0037 099 099800 1286 106

I'm in the process of tweaking this list a bit to see if more candidate puzzles can be generated...

tarek
[Edit 12.04.07 0830: Added some ERs]
[Edit 13.04.07 0830: ERs provided for all puzzles]
[Edit 13.04.07 2045: Removed last column in table]

[tarek]

Here is another list....ERs will be updated when obtained...


Puzzle_________________________________________________________________________________________ #__________ Q2 Qhdst_ QhdstX Sxr9_ ER
000070100000008050020900003530000000062000004094600000000001800300200009000050070 ULTRA0100 106 99992 99992 1331 109
007002000500090400010600000400050003060100000002007000000000810900000306000080059 ULTRA0101 103 99991 99991 1069 107
070200009003060000400008000020900010800004000006030000000000600090000051000700002 ULTRA0102 018 99874 99884 1188 107
400000009080002060001000700300074000000805000000016050007000100020600080900000004 ULTRA0103 011 99873 99873 1004 105
100080000005900000070002000009500040800010000060007200000000710000004603030000402 ULTRA0104 010 99865 99875 1009 107
000007090000800400003060001420010000031000002605000000060400800500020006000009070 ULTRA0105 007 99864 99884 1393 106
900000004010006020005000800300080000000107000000635070008000500060200010400000009 ULTRA0106 016 99862 99882 1063 104
100009080030600001007020000000000039400000500000005810900008100002070000060300000 ULTRA0107 007 99861 99861 1326 107
097000000301005000045000800003008400000020060000100009700004300000900001000060020 ULTRA0108 013 99852 99864 1030 105
000007080030500100000020009100300500600090002000008070360000000007000000451000600 ULTRA0109 124 99844 99874 1039 104
400002003060000010008000500700604000000070004000350000005000800017000060300900007 ULTRA0110 101 99844 99874 1011 104
000900100000080007004002050200005040000100900000070008605000030300006000070000006 ULTRA0111 088 99844 99864 1005 107
000600001000020400300009050090005030000040200000100006570008000002000000080000090 ULTRA0112 018 99843 99863 1103 106
007580000000030000000076005400000020090000100003060008010600900006800003200000040 ULTRA0113 006 99843 99863 1074 105
000001080030500200000040006200300500000008010000060004050000700300970000062000000 ULTRA0114 008 99842 99862 1263 107
010000009005080700300700060004250000000000000000840200008007500600000030090000001 ULTRA0115 008 99842 99852 1048 105
000900060010003400003070002040000000008000500051008000000600090000020007300004100 ULTRA0116 008 99842 99852 1026 105
200000006050008070004000100900803000000040000000975030001000400080700050600000002 ULTRA0117 017 99842 99842 1012 104
006003000900080200070400000003006000040700000800020090500000008000000709000510020 ULTRA0118 008 99835 99886 1291 106
700040000008300009050006000003800000400070000010005600000000540000009102020000906 ULTRA0119 007 99834 99857 1112 105
600005020040700000009080000010000302000000087000200104070400003500006000008090000 ULTRA0120 008 99833 99853 1209 106
010300000000009000000710050004050900200000006070800030600000002080030070009000400 ULTRA0121 109 99833 99883 1136 106
000008070000300200005040009260094000059000006401000000000200300100060004000007080 ULTRA0122 007 99831 99896 1332 106
003700000050004000100020080900000012000000400080010090007300000200090006040005000 ULTRA0123 116 99831 99862 1122 105
000000100600000874000007026030400000005090000100008002009050000200001008040300000 ULTRA0124 109 99828 99828 1825 105
300000007010008020004000900600701000000582000000063050009000400080200010700000003 ULTRA0125 018 99811 99831 1225 105
000800300000010005004002070200007040000300807000050001907000060600009000050000000 ULTRA0126 094 99800 99800 1201 106


tarek

[Edit 13.04.07 2050: Removed last column in table]
[Edit 14.04.07 2325: Updated ERs]
[Edit 15.04.07 1700: Missing header row returned]

[AW]

Lots of impressive puzzles have been found recently!

I've run the top ones through my solver to see if any could score some jellies, and the top tier now has three additional members! Solver even says there's a new champion that beat dml 1/07... one that somehow manages to require three jelly runs to break!

Runners up are now :
Coloin St.Patrick 1 : 5 swords & 1 jelly
Tarek 04/08/2 : 8 swords & 1 jelly
Ocean's New Year's present for RW : 13 swords & 1 jelly
dml 1/07 : 17 swords & 1 jelly

And the monstrous new champion, the very first puzzle to score more than one jelly, the one, the only...

JPF 04/07/1 : 12 swords & 3 jellies
100000002090400050006000700050903000000070000000850040700000600030009080002000001

Never thought the back-door conjecture would be broken, never thought the single jelly conjecture would be broken either. But here's a monster to break both!

Nice work!

[udosuk]

AW, could you please show us a link explaining your "jelly runs"?

And an innocent little suggestion: aside from "JPF 04/07/1" this puzzle could have a nickname "Easter Monster"!

[coloin]

I like the "Easter Monster"

However I have just found out that I cant solve by hand ANY of the puzzles I make - even the easiest with suxrat9 of 90 !

I found the relevant post called Looking for inspiration on implication:

Thankyou AW

The program appears to deliver - is it available - usable ?


Tarek - Can it solve [the 18 week] waiting list crisis ?

I think the main reason we have found the difficult puzzles is that we have generated so many - however there could easily be a harder one buried in my suexrat data !

I searched these two batches of 16-bases... - no hard puzzles at all !

Code: Select all

1.......2.9.4...5...6...7...5.......................4.7.....3...4...9.8...2.....1
6.......2.9.4...5...1...7...5.......................4.7.....3...4...9.8...2.....1
1.......7.9.4...5...6...2...5.......................4.7.....3...4...9.8...2.....1
6.......2.9.4...5...1...7...5.......................4.2.....3...4...9.8...7.....1

..1.....2.9.4...5.6.....7...5.......................4.7.....3...4...9.8...2.....1
..6.....2.9.4...5.1.....7...5.......................4.7.....3...4...9.8...2.....1
..1.....7.9.4...5.6.....2...5.......................4.7.....3...4...9.8...2.....1
..6.....2.9.4...5.1.....7...5.......................4.2.....3...4...9.8...7.....1

702977 sol.
695087 sol.
705798 sol.
700225 sol.

237493 sol.
240345 sol.
233958 sol.
239572 sol.

Tarek - all your puzzles in your ULTRA0000 - ULTRA0037 appear to be with this equiv 16-base !

Code: Select all

1.......9.3...4.7...6...2...5.......................8...2...6...4.7...3.9.......1
537480 sol.

There are many of these 16-bases could we document which ones that we feel we have fully searched .

I think JPF may be finding difficult puzzles from other puzzles and therefore these bases havent been fully searched.

C

[AW]

My solver uses an extended form of "tabling", iteratively applying a small set of heuristics. More advanced heuristics are only applied when the program can no longer advance using the easier ones. A "sword" means the program had to check for pair eliminations based on two constraints (second to last pattern). A "jelly" mean it had to resort to checking its last and most complicated heuristic, pair eliminations based on three constraints.

There was a discussion about the approach here :
http://forum.enjoysudoku.com/viewtopic.php?t=5277

As a basis for comparison, the AI Ertana does not score a sword. Most of the really really hard puzzles mentioned here and in the "hardest" thread get one sword. Some get more than one and, so far, only the five puzzles mentioned in the previous post require a jelly. JPF's is the only one for which solver got stuck again after the first jelly.

[RW]

There are many of these 16-bases could we document which ones that we feel we have fully searched .

Seems like a good idea to me. I've done full searches on these three 16 clue bases:

Code: Select all

1.......6.2.8...5...3...4...8.......................9...4...1...7...9.2.6.......3
681116 solutions
7....9..2.6.....3...5...1..........94...................1...5...3.....6.2...4...7
307360 solutions
1.......6.2.9...5...3...4...9..............8............5...3..4...8..2..6......7
8688804 solutions

The third was a complete waste of time, produced <50 puzzles. I've reported all puzzles with gsfr>99800 from the two first grids in this thread.

I'm wondering how hard puzzles are of interest in these times with so many SE 10.5-10.9 puzzles around. If we want to focus on collecting more 11+ puzzles, then standard gsfr is a good screening tool. All known 11+ puzzles have gsfr>99990.

However I have just found out that I cant solve by hand ANY of the puzzles I make - even the easiest with suxrat9 of 90 !

Did you find a puzzle this easy using this method? Could you post it please? I've been doing some thinking on why this method fails to produce easy puzzles, maybe your puzzle proves my theories wrong...

RW

[tarek]

Tarek - all your puzzles in your ULTRA0000 - ULTRA0037 appear to be with this equiv 16-base !

Code: Select all

1.......9.3...4.7...6...2...5.......................8...2...6...4.7...3.9.......1
537480 sol.

There are many of these 16-bases could we document which ones that we feel we have fully searched .

true.... the 2nd list is not.... My next list is a combination of fully searched 3 16-base puzzles....

I usually longlist using Q2 (>5) then the final shortlist (gsfr>9800, Suexr9>1000).

All the 16 base puzzles that I start are fully searched, I will post them after my next list (which has a suexrat9 puzzle of >2000!!!)

tarek

[coloin]

AW's rating program certainly has the right "attitude".

RW a quick look found the puzzle - as you can see the first doesnt fit the pattern - by mistake - I did a whole template with 2 clues in box 8 ! So that explains that !

Code: Select all

rating:     89 ,  1.......7.9.4...5...6...2...5..3........72.......6..4...7...3...4...9.8.....2...1  - 7.1 [easiest]
rating:    955 ,  6.......2.9.4...5...1...7...5..1.......3........968.4...2...3...4...9.8.....7...1  - 9.8 [hardest]

Code: Select all

+---+---+---+
|1..|...|..7|
|.9.|4..|.5.|
|..6|...|2..|
+---+---+---+
|.5.|.3.|...|
|...|.72|...|
|...|.6.|.4.|
+---+---+---+
|..7|...|3..|
|.4.|..9|.8.|
|...|.2.|..1|
+---+---+---+  7.1

heres the "easiest" genuine 16 clue template puzzle out of my last run. If I get a less good one I will post it .

Code: Select all

rating:    119 ,  6.......2.9.4...5...1...7...5...3......127........8.4.7.....3...4...9.8...2.....1  -  7.2
+---+---+---+
|6..|...|..2|
|.9.|4..|.5.|
|..1|...|7..|
+---+---+---+
|.5.|..3|...|
|...|127|...|
|...|..8|.4.|
+---+---+---+
|7..|...|3..|
|.4.|..9|.8.|
|..2|...|..1|
+---+---+---+ 7.2

I think I know why we are able to make these puzzles though .....

The grid solution rate of these 16 clue templates is abnormally low for only 16 clues.
Normally in low clue puzzles the number of grid solutions for a subgrid is reduced becaues clues are insertable by singles etc.
This is not the case with our sub grids- there are sometimes no insertable clues even at the 16+3 clue stage.

So why is there a low solution count for these 16 clue subgrids ? Some process has to be at work here.

Well this count is dependant on the unavoidables........if there are more unavoidables and their iterations [valancy by Red Ed] then there will be more grid solutions......

.......But if you have a big [minimal] unavoidable you cannot have a smaller unavoidable within this - therefore the grid solution rate is reduced.

This didnt happen with our 17 clue puzzle megaclue puzzle - one of its cluues was in a 55 clue unavoidable set out of 65 [81-16] possible clues. Having said that the grid wasnt difficult ! SE 3.0 But it is all the clues which count of course !

Unless of course there are bigger unavoidables than 55 in our grids ?

Eg a 56 clue with only 5 clues not involved........this would be unlikely

If the grid solution rate is reduced then you probably have a chance to complete the puzzle in less clues. Also if you only have big unavoidables left there is also a bigger chance that you can solve the grid with fewer remaining clues. The lower solution rate means we can find more grids which solve with a full box 5 - the fact that we are able to easily find all the nonminimal grids with full central boxes which complete means that we can perform a complete and extensive search - we will always be able to remove clues from the full box. It seems that 4 or 5 clues in the central box is optimal - but I suspect that this is the case because many more puzzles are generated which increases the chances of finding a "monster".

I wonder if we could find a 3 clue diagonal in box 5 which completes whether this would be the "ultimate" .

Code: Select all

+---+---+---+
|x..|...|..x|
|.x.|x..|.x.|
|..x|...|x..|
+---+---+---+
|.x.|..x|...|
|...|.x.|...|
|...|x..|.x.|
+---+---+---+
|x..|...|x..|
|.x.|..x|.x.|
|..x|...|..x|
+---+---+---+

Its existance as a hard puzzle might be self determinimng [if it exists]
i.e the grid solution count of the 16 clue base will be lowish - to be solvable with 3 clues, You will only get this lowish rate if it has non similar clues in a band. There wil not be very many puzzles of this pattern - so that goes against it being super hard - but the absence of a clue does might just make up for everything.

I also wonder whether there is a relationship between the "hardness" and the size of the the largest unavoidable uniquely associated with each clue. Or maybe the its do to with the size of the largest unavoidable in the weakest [least grid sol.] clue.

Maybe our method makes puzzles with clues which each has an associated large unavoidable............

I will publish my waiting list - I only hope its not the same as tarek's !

C

[tarek]

here is my 3rd list (ERs will follow)

Puzzle_________________________________________________________________________________________ #__________ Q2_ Qhdst Sxr9_ ER
200000006070009080004000300000095000050607010000108000003000400090800070600000002 ULTRA0200 119 99992 1455 108
800000007040001030009000600000532000050108020000400000006000900010300040700000008 ULTRA0201 124 99992 1380 106
800000005040003020007000100000004000090702060000639000001000700030200040500000008 ULTRA0202 119 99992 1337 107
200000006050080010004000900070301000000820000000705030009000400080010050600000002 ULTRA0203 087 99991 2105 114
900000001030004070006000200050302000000060000000078050002000600040700030100000009 ULTRA0204 147 99892 1156 107
800000005040003020007000100000539000090004060000002000001000700030200040500000008 ULTRA0205 122 99873 1165 106
700000003060002010008000500040903000000050000000061040005000800020100060300000007 ULTRA0206 138 99865 1246 106
800000009040001030007000600000023000050904020000105000006000700010300040900000008 ULTRA0207 117 99863 1454 108
100000009040008030002000600070034000000702000000850070006000200080300040900000001 ULTRA0208 138 99857 1611 108
800000005060003010007000400090021000000409000000630090004000700030100060500000008 ULTRA0209 127 99855 1634 108
500000009030007040001000200080601000000020000000930080002000100070400030900000005 ULTRA0210 163 99855 1061 106
400000008020007060009000300000275000010390050000016000003000900070600020800000004 ULTRA0211 108 99854 1220 104
300000009020001070005000800000002000040087060000614000008000500010700020900000003 ULTRA0212 111 99853 1182 105
100000009060007050002000400030850000000360000000701030004000200070500060900000001 ULTRA0213 140 99847 1214 107
700000003010004020008000500060040000000031000000902060005000800040200010300000007 ULTRA0214 132 99846 1359 106
200000006090005010004000300080009000000820000000573080003000400050100090600000002 ULTRA0215 147 99846 1060 106
200000006050008090001000300070485000000709000000020070003000100080900050600000002 ULTRA0216 141 99845 1051 104
800000005040001030007000900020516000000020000000840020009000700010300040500000008 ULTRA0217 011 99843 1139 104
100000004070006020005000800030029000000080000000705030008000500060200070400000001 ULTRA0218 144 99843 1131 105
900000007030008040006000200010389000000010000000205010002000600080400030700000009 ULTRA0219 151 99841 1344 106
700000003010004020008000500060902000000630000000001060005000800040200010300000007 ULTRA0220 127 99837 1262 106
300000009010006050002000400070060000000701000000845070004000200060500010900000003 ULTRA0221 139 99835 1491 106
400000007050009080002000600030100000000020000000458030006000200090800050700000004 ULTRA0222 152 99834 1218 105

tarek

[Edit 15.04.07 1700: ERs updated]

[JPF]

OK
Let's go !

Puzzle_________________________________________________________________________________________*____________*gsfr* Sxr9
100000002030400050006000700050804000000073000000900080700000600040008090002000001 JPF 04/14/01 99995 1567
100000002030400050006000700050803000000074000000900080700000600090008030002000001 JPF 04/14/02 99995 1453
100000002030400050006000700080309000000070000000500040700000600040003080002000001 JPF 04/14/03 99994 1419
100000002030400050006000700080309000000070000000500040700000600050003080002000001 JPF 04/14/04 99994 1396
100000002003400050060000700000308040000006000008540000020000100700000006005090030 JPF 04/14/05 99992 1655
100000002003400050060000700000045080000906000004380000020000100700000006005030090 JPF 04/14/06 99992 1600
100000002003400050060000700000089040000506000008340000020000100700000006005090030 JPF 04/14/07 99992 1540
100000002003400050060000700000890040000306000009040000020000100700000006005080030 JPF 04/14/08 99992 1504
100000002030400050006000700040803000000070000000950030700000600090008040002000001 JPF 04/14/09 99992 1495
100000002003400050060000700000030080000506000009840000020000100700000006008090030 JPF 04/14/10 99992 1482
100000002003400050060000700000035040000806000008940000020000100700000006005090080 JPF 04/14/11 99992 1456
100000002030400050006000700080309000000060000000500040700000600040003080002000001 JPF 04/14/12 99992 1437
100000002003400050060000700000040080000506000009830000020000100700000006008090030 JPF 04/14/13 99992 1426
100000002003400050060000700000030040000506000008940000020000100700000006005080090 JPF 04/14/14 99992 1394
100000002030400050006000700050809000000010000000057040700000600040008090002000001 JPF 04/14/15 99992 1159
100000002003400050060000700000230040000006000008590000020000100700000006005080090 JPF 04/14/16 99992 939
100000002030400050006000700080309000000070000000501040700000600090008030002000001 JPF 04/14/17 99992 849
100000002003400050060000700000320040000006000008940000020000100700000006005080090 JPF 04/14/18 99992 844
100000002030400050006000700080905000000010000000307040700000600050008090002000001 JPF 04/14/19 99992 820
100000002030400050006000700080309000000070000000108040700000600090005030002000001 JPF 04/14/20 99992 739
100000002003400050060000700000803040000006000008540000020000100700000006005090080 JPF 04/14/21 99991 1892
100000002003400050060000700000080040000506000009340000020000100700000006005090080 JPF 04/14/22 99991 1383
100000002003400050060000700000839040000620000000047000020000600700000001005090030 JPF 04/14/23 99884 561
100000002030400050006000700050306000000074000000150080700000600040005090002000001 JPF 04/14/24 99882 719
100000002030400050006000700080092000000604000000850040200000600040003080007000001 JPF 04/14/25 99871 941
100000002030400050006000700050800000000379000000051080700000600090005030002000001 JPF 04/14/26 99854 856
100000002030400050006000700050802000000079000000350090700000100090008030002000006 JPF 04/14/27 99852 1051
100000002030400050006000700050004000000370000000251080700000600080009030002000001 JPF 04/14/28 99852 706
100000002003400050060000700000583040000006000008040000020000100700000006005090030 JPF 04/14/29 99851 1207
100000002003400050060000700000589040000306000008040000020000100700000006005090030 JPF 04/14/30 99851 996
100000002003400050060000700000378040000006000008940000020000100700000006005080090 JPF 04/14/31 99851 653
100000002003400050060000700000730040000806000008540000020000100700000006004080090 JPF 04/14/32 99850 858
100000002030400050006000700080904000000073000000081040700000600050009030002000001 JPF 04/14/33 99843 1007
100000002003400050060000700000230040000010000008549000020000100700000006005090080 JPF 04/14/34 99843 993
100000002003400050060000700000180090000006000004900000020000100700000006005040030 JPF 04/14/35 99842 703
100000002030400050006000700050300000000670000000081040700000600090008030002000001 JPF 04/14/36 99842 457
100000002030400050006000700050028000000704000000351090700000600040003080002000001 JPF 04/14/37 99841 768
100000002030400050006000700050809000000075000000300080700000600040008030002000001 JPF 04/14/38 99836 1128
100000002030400050006000700080300000000960000000851040700000900050003080002000001 JPF 04/14/39 99836 998
100000002003400050060000700000038040000540000008907000020000100700000006005080030 JPF 04/14/40 99834 940
100000002030400050006000700080024000000105000000890030700000100050003080002000006 JPF 04/14/41 99834 685
100000002003400050060000700000540030000806000009030000020000100700000006005090080 JPF 04/14/42 99833 1204
100000002030400050006000700050008000000360000000901040200000600040003090007000001 JPF 04/14/43 99833 1119
100000002030400050006000700080320000000604000000851040900000600050003080002000001 JPF 04/14/44 99833 932
100000002003400050060000700000814090000320000004509000020000100700000006005000030 JPF 04/14/45 99832 881
100000002003400050060000700000018040000306000005940000020000100700000006009080030 JPF 04/14/46 99832 849
100000002030400050006000700080059000000070000000300040700000600040005030002000001 JPF 04/14/47 99832 815
100000002030400050006000700050840000000079000000301080700000600040008030002000001 JPF 04/14/48 99831 894
100000002003400050060000700000100040000285000009007000020000800700000006005090030 JPF 04/14/49 99831 776
100000002003400050060000700000130040000806000005900000020000100700000006004090030 JPF 04/14/50 99829 1026
100000002003400050060000700000378040000001000008540000020000600700000001005080090 JPF 04/14/51 99828 740
100000002003400050060000700000080040000506000009340000020000100700000006005030090 JPF 04/14/52 99827 1086
100000002003400050060000700000300040000806000009540000020000100700000006005080030 JPF 04/14/53 99826 1302
100000002030400050006000700050800000000079000000350080700000600040008030002000001 JPF 04/14/54 99826 1056
100000002030400050006000700080009000000604000000831040700000600050003080002000001 JPF 04/14/55 99826 861
100000002003400050060000700000308040000006000008940000020000100700000006005030080 JPF 04/14/56 99825 982
100000002030400050006000700080309000000070000000500040700000600040008030002000001 JPF 04/14/57 99824 1124
100000002003400050060000700000000040000826000008507000020000900700000001009030080 JPF 04/14/58 99824 1051
100000002030400050002000600050708000000290000000301040900000800040007030006000001 JPF 04/14/59 99824 838
100000002030400050006000700040028000000690000000301080700000600080005030002000001 JPF 04/14/60 99824 796
100000002003400050060000700000080040000096000008540000070000100200000006005030080 JPF 04/14/61 99824 795
100000002003400050060000700000800040000306000009540000020000100700000006005080030 JPF 04/14/62 99823 1200
100000002003400050060000700000053040000806000009040000020000100700000006005090080 JPF 04/14/63 99823 1051
100000002003400050060000700000130040000806000008570000020000100700000006005090080 JPF 04/14/64 99822 682
100000002003400050060000700000068040000392000009500000020000100700000006005080030 JPF 04/14/65 99821 1074
100000002003400050060000700000300040000806000005940000020000100700000006009030080 JPF 04/14/66 99820 911
100000002003400050060000700000108040000036000008940000020000100700000006009050080 JPF 04/14/67 99818 1079
100000002003400050060000700000150040000806000009040000020000100700000006005090030 JPF 04/14/68 99800 786

JPF

[RW]

I think I know why we are able to make these puzzles though .....

I was attacking this problem from a more practical point of view. My main question was "why can't we make these puzzles easy?"

Say we have a grid like this:

Code: Select all

 *-----------*
 |X..|...|..X|
 |.X.|X..|.X.|
 |..X|...|X..|
 |---+---+---|
 |.X.|OOO|...|
 |...|OOO|...|
 |...|OOO|.X.|
 |---+---+---|
 |..X|...|X..|
 |.X.|..X|.X.|
 |X..|...|..X|
 *-----------*

All of box 5 is solved and then we have the 16 clue template with no digit appearing twice in the same band or stack. First of all, there cannot be any direct singles, neither hidden or naked. Eliminations from locked candidates are also impossible. Bivalue cells may only exist in r123c6, r4c789, r6c123 or r789c4, if the three digits they see in box 5 are different than the four clues they see from the 16 clue template. There can be at most one bivalue cell in each of these four sets, which means that no two bivalue cells can see each other. This renders XY-wings and longer XY-chains unusable. Even though there might be two bivalue cells in the same band/stack, these cannot possibly have the same two values, which renders standard URs unusable.

Subsets are possible:

Code: Select all

 *-----------*
 |X..|...|..X|
 |.X.|B..|.A.|
 |..X|...|X..|
 |---+---+---|
 |.B.|OOO|...|
 |...|OOO|...|
 |...|OOO|.X.|
 |---+---+---|
 |..X|...|X..|
 |.A.|..X|.X.|
 |X..|...|..X|
 *-----------*

where a hidden pair is formed in one of the corner boxes. This can be avoided by choosing a 16 clue template that doesn't cause any hidden pairs. These hidden pairs are not found in any of the hardest puzzles created in this thread. Apart from this I don't think any subsets are possible.

I haven't seen any nishio eliminations either in a grid like this, but I can't prove they are impossible. With a slight alteration of the 16 clue template some fishy eliminations can exist. Example from the "original rare shape"-thread:

Code: Select all

 *-----------*
 |1..|.2.|..3|
 |.4.|...|.5.|
 |..6|...|7..|
 |---+---+---|
 |...|894|...|
 |7..|162|..9|
 |...|375|...|
 |---+---+---|
 |..1|...|4..|
 |.5.|...|.6.|
 |3..|.1.|..2|
 *-----------*

Has two swordfishes. Note that both of these rely on the fact that the clues in box 2&8 and 4&6 are in the same row/column. If they were not (which they aren't in any of the hardest puzzles) then swordfishes like these cannot exist.

At this point we are left with quite few techniques at our disposal. In SE there really is no other option than forcing chains. In general there are no short forcing chains. There is very few bivalue cells, so they must mostly rely on bilocation units. This brings me to another thing I just noticed.

I searched these two batches of 16-bases... - no hard puzzles at all !

I think I know why. You have three 4's in all of them. If a digit exists only twice in the 16 clue template, then there might be only two boxes, in opposite corners of the puzzle, where the digit appears as a candidate in only two cells. If the 16 clue base is chosen wisely, then it can be made so that the base in itself doesn't contain more than one bilocation unit for each digit. JPF's easter monster came from a perfect base in this sence:

Code: Select all

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

No more than one bilocation unit per digit. This base that I searched earlier is a total disaster:

Code: Select all

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

Two bilocation units for digits 1,2,3,5,6,7. Not suprisingly this base didn't reveal a single puzzle with ER>10.6 or gsfr>99900.

Coloin, in your 16 clue templates with three 4's there's immediately three bilocation units for digit 4, which will make it easier to find short forcing chains. Also, all of your templates have the hidden 45 pair in box 1.

So here's my advice on creating good 16 clue templates:
-No digit appears more than once in any band/stack
-No digit appears more than twice in the template
-No direct hidden pairs in any of the corner boxes
-No digit has more than one bilocation unit

RW