Fully symmetrical puzzles

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

Postby Havard » Tue May 09, 2006 1:19 pm

I ran my solver through most of these (1260), and found among other interesting stuff that these puzzles has an abnormal amount of single-digit eliminations! (like strong links, fishes, coloring etc)

I counted 5607 Box-Line / Line-Box eliminations, and 6548 eliminations of type x-wing, swordfish, 2/3/4 strong links, finned and franken variations etc. For 1260 puzzles that means quite a few of these per puzzle. I guess the extreme symmetry of these puzzle might be a possible cause of this?

Havard
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Postby gsf » Tue May 09, 2006 1:44 pm

JPF wrote:I’ve a few questions :
a) Any new patterns ?
b) I assume that all your puzzles are non-equivalent. Right ?
I would be interested to know how you test that.

there are 65 isomorphic dups in ocean's post
the first puzzle is the canonical form used
(min row order lexicographic with top left box 123/456/789)
followed by the equivalent puzzles
in this case all dups were doubles
Code: Select all
# 1-4-4-2-0
000000009450109000080006000070005000530704000000000008000000490000000530002070000
120000034300000002000050000000102000006000300000704000000070000500000008240000097
120000034500000006000010000000103000007000800000904000000060000400000007370000049

# 2-4-1-0-0
000000680050100007009003000090700003000000420008002000600040000800070000002008000
000102000001030400040000050300000001060000070800000002070000090003010600000708000
000102000003040500010000060200000007050000010300000004080000090004030800000407000

# 4-9-0-2-0
000007600000100020009030004000300070005040002000006100360000000570000000000020008
010000020300000004004050600000102000007000800000804000006090500500000001020000030
010000020300000004004050600000203000007000800000907000006040900900000001020000030

# 4-9-0-2-0
000007600400080003000200040068000000071000000000030002500040008000001700000600090
010000020300000004005010600000306000002000500000407000006080100400000007090000030
010000020300000004005030600000106000004000500000207000006080300900000001020000070

# 4-9-0-2-0
000007600400080003000200040200040008000001700000600030000090005071000000068000000
010000020300000004002050600000304000001000700000608000006090100400000008070000030
010000020300000004004050600000102000003000700000608000006090300700000001020000080

# 1-8-6-0-0
000050000050109030700000400200000800030504090000010000008000002040000070090000050
000010000020000030001405600003000200700000008004000100006301400080000070000050000
000010000020000030004506700003000400800000005005000800007403500010000060000020000

# 4-9-0-2-0
000050600400009003000200010000006008075000000091000000300008004000900020000010500
010000020300000004002050600000207000008000900000403000009060800500000003040000010
010000020300000004004050600000102000007000800000904000008060700200000003050000010

# 4-9-0-2-0
000057000400000200000036000200900500005000006010000070007000008060000010900500300
010000020300000004002050600000107000006000800000302000007020500400000001090000030
010000020300000004004050600000104000006000700000308000008040500900000001020000030

# 1-4-4-2-0
000400080050000000000103046201000000000060200804000000000804027000300090070000000
120000034300000002000050000000102000006000700000804000000080000500000007240000098
120000034500000006000010000000103000005000700000804000000060000400000009390000048

# 4-9-0-2-0
000400080400009200000060001000300060500002400000010003000008900017000000062000000
010000020300000004005010600000306000002000500000407000006080100400000009090000030
010000020300000004005030600000106000004000500000207000006080300900000001020000090

# 1-8-6-0-0
000400600000080030000600500070000000000002001805300200010000000602800900000005007
000010000020000030001405600003000200700000008004000100009204300080000070000060000
000010000020000030004506700003000500800000007005000900006703800010000060000020000

# 1-8-6-0-0
000400600050000000709060001000003010000090002000020007601070008000500900040000000
000010000020000030001405600003000100700000008004000200002503400080000070000090000
000010000020000030003405600004000700800000006007000500005603900010000040000020000

# 1-4-4-2-0
000406709050000000000002004204000000807000000000060030000008006090000000000304208
120000034300000005000060000000207000008000400000305000000010000400000007750000083
120000034400000005000010000000203000006000700000408000000050000800000002740000013

# 4-9-0-2-0
000407000000109000700000005004000010500030008090000500001000090800020007060000400
010000020300000004002050600000104000007000200000703000006080700400000001090000030
010000020300000004004050600000107000007000400000203000006080700900000001020000030

# 4-9-0-2-0
000407000000109000700000005004000010500030008090000700001000090800020007060000400
010000020300000004002050600000104000007000800000203000006020500400000001090000030
010000020300000004004050600000104000007000800000203000006040500900000001020000030

# 4-9-0-2-0
003000009400000020080100500060000300000028000000047000002000007070500800900000040
010000020300000004002050600000106000007000800000402000006090500400000001080000030
010000020300000004004050600000204000007000800000306000006090500800000001020000030

# 4-9-0-2-0
003000080050009100700000006008000070010008900600000003040000500000630000000170000
010000020300000004005010600000205000007000800000403000008060500200000003040000010
010000020300000004005030600000102000007000800000504000008060500200000003040000010

# 1-8-6-0-0
003000700400000001009000500004013900070000020000600000005074100010000060000200000
000010000020000030001405600003000200700000008004000100005206300080000070000030000
000010000020000030001405600004000100300000002007000800005806700010000040000070000

# 0-4-2-1-0
003007600450000000009006100240500010000000000000001300000003800000000000890200050
000000000001203400020000050060030070000801000070050090050000060006408100000000000
000000000001203400040000050060050070000102000070080040050000090003809100000000000

# 4-9-0-2-0
003050009400000030080000600005040001600000040030000800000805000000601000002000007
010000020300000004002050600000104000006000300000607000008090700200000003040000010
010000020300000004004050600000105000007000800000209000005030900200000003040000010

# 4-9-0-2-0
003050009400100000080006000060004000007090005900800000000000160000000340002010000
010000020300000004002050600000307000008000900000408000005020700400000001080000030
010000020300000004005010600000107000006000500000304000007080900200000003040000060

# 4-9-0-2-0
003050700400009000080100000010600000005070300900008000000000041002060000000000058
010000020300000004005010600000203000007000500000806000006090700400000008020000030
010000020300000004005030600000104000007000500000608000006090700400000001020000080

# 4-9-0-2-0
003400000050080000700002006200005000000000390000000410040060000600001007005300000
010000020200000003004050600000103000006000400000708000005020800900000006030000010
010000020200000003004050600000103000006000400000708000007020500300000001040000090

# 4-9-0-2-0
003400000050080000700002006200005000000000490000000310040060000600001007005300000
010000020200000003004050600000103000006000400000708000005020700900000006030000010
010000020200000003004050600000103000006000400000708000008020500300000001040000090

# 4-9-0-2-0
020000080006000100700003005000910000500000002000640000300007004010000090002000600
010000020200000003004050600000107000006000400000803000005020800900000007030000010
010000020200000003004050600000107000006000400000803000007020500300000001080000090

# 4-9-0-2-0
020000600400080007009000050010000900005000020800070001300000006000301000000205000
010000020300000004004050600000102000007000800000804000005060900700000001020000030
010000020300000004005010600000306000007000500000408000009080700200000003040000060

# 4-9-0-2-0
020006000400700000009030005001040000000000960000000820070002000600300000005090003
010000020300000004002050600000304000007000200000608000006090700400000001050000030
010000020300000004004050600000102000007000400000608000006090700500000001020000030

# 4-9-0-2-0
020006009000080100000200040000040800000100050090002006508000000601000000000007003
010000020300000004002050600000104000007000200000608000006090700400000001080000030
010000020300000004004050600000203000007000400000806000006090700800000001020000030

# 1-8-6-0-0
020007000400100000080006000000000300001040000060003028000000800007020000030008056
000010000020000030001405600003000200700000008004000100006102400080000070000060000
000010000020000030001405600006000400300000002007000500004608100050000070000040000

# 1-8-6-0-0
020007000400100000080006000000000300001040000060003028000000800007020000030008065
000010000020000030001405600003000200700000008004000100006201400080000070000060000
000010000020000030001405600006000400300000002007000500004806100050000070000040000

# 1-8-6-0-0
020007000400100000080006000000000300001040000070003028000000800007020000030008065
000010000020000030001405600003000200700000008004000100006201300080000070000060000
000010000020000030001405600006000400300000002007000500005708100040000060000050000

# 1-8-6-0-0
020007000400100000080006000001040000000000900060002038000000800007020000030008056
000010000020000030001405600003000200700000008004000100002106400080000070000090000
000010000020000030003405600006000400700000006008000900004603800010000050000020000

# 1-8-6-0-0
020007000400100000080006000001040000000000900060002038000000800007020000030008065
000010000020000030001405600003000200700000008004000100002601400080000070000090000
000010000020000030003405600006000700800000006009000400004603900010000050000020000

# 1-8-6-0-0
020007000400100000080006000001040000000000900070002038000000800007020000030008056
000010000020000030001405600003000200700000008004000100002106300080000070000090000
000010000020000030003405600006000400700000006008000900005603800010000050000020000

# 1-8-6-0-0
020007000400100000080006000001040000000000900070002038000000800007020000030008065
000010000020000030001405600003000200700000008004000100002601300080000070000090000
000010000020000030003405600006000700800000006009000400005603900010000050000020000

# 4-9-0-2-0
020007080400080000009300000000000403000000802070001000300040000008900000010002050
010000020300000004005010600000106000007000500000203000006080700400000001020000030
010000020300000004005030600000104000007000500000603000006080700400000001020000030

# 4-9-0-2-0
020007600400080000009300000008900000300040000060002900000000028030005000000000014
010000020300000004005010600000203000004000500000706000006080900400000001020000030
010000020300000004005030600000104000002000500000607000006080900400000001020000030

# 4-9-0-2-0
020400000000000037000000015005010000070900800600003000040800900800005000001060000
010000020300000004002050600000104000006000700000809000005020800400000001090000030
010000020300000004004050600000203000006000700000809000005040900800000001020000030

# 4-9-0-2-0
020400000000000073000000051001060000070900800900005000005010000600003000040700300
010000020300000004004050600000203000007000800000906000006090700900000001020000030
010000020300000004005030600000102000006000700000305000008090500200000008040000010

# 4-9-0-2-0
020400009000080070000003500000600007601000000805000000000005800040700001000010030
010000020300000004002050600000207000006000700000403000008090500400000001060000030
010000020300000004004050600000102000006000700000704000008090500600000001020000030

# 4-9-0-2-0
020400009006080000700002000000000750040300000000000820300006000008070000090500001
010000020300000004002050600000107000006000800000402000005090700400000001080000030
010000020300000004004050600000204000006000700000308000005090800700000001020000030

# 0-4-2-1-0
020450009006000030000000000007001060000240000008005010070890004000000000001000070
000000000001203400020000050030050060000107000080060070060000080002401900000000000
000000000001203400040000050050060020000701000080050040090000080003106700000000000

# 1-4-4-2-0
023000000000100070089000000200000000000003400000045308000008900600000000000031504
120000034300000001000050000000105000006000700000802000000090000800000002430000017
120000034300000005000010000000206000007000600000803000000090000500000002480000053

# 1-4-4-2-0
023406000006700000000000500000000067000000021900010000071608000000000300004200000
120000034300000005000060000000207000008000700000305000000010000400000002750000093
120000034400000005000060000000206000007000800000403000000090000300000002840000016

# 4-9-0-2-0
100000009006700020080000500200000001090000800007600030000014000000093000002000050
010000020300000004005010600000203000006000500000708000008090100400000009020000030
010000020300000004005030600000104000006000500000708000007090300400000001020000090

# 4-9-0-2-0
100000080050000002009030600030000006007020300800000050000805000004000900000107000
010000020300000004002050600000307000006000100000408000005090800700000003040000060
010000020300000004004050600000107000006000300000208000005090800200000006070000010

# 1-4-4-2-0
100007000000000023000000054034690000000000900008020000000000800045730000007040000
120000034300000002000050000000102000006000300000704000000080000500000007240000098
120000034500000006000010000000305000007000800000409000000060000400000007370000049

# 1-4-4-2-0
100007000000000230000000540000000006030010000045920000000000008014630000060040000
120000034300000002000050000000106000007000400000308000000020000400000001690000083
120000034400000005000010000000304000006000200000705000000080000200000009340000017

# 4-9-0-2-0
100007600050000030009000005000930000000840000500000100030000090800006700005000002
010000020200000003004020500000106000007000800000304000008050700900000002030000010
010000020200000003004050600000103000006000200000708000005040800900000002030000010

# 1-8-6-0-0
100050000006009000700030000200010305090600000000000040500020108030700000000000050
000010000020000030001405600003000200700000008004000100006201400080000070000090000
000010000020000030004506700003000500800000007007000900005703400010000060000020000

# 1-8-6-0-0
100050000006009000700030000200010305090600000000000040500020801030700000000000050
000010000020000030001405600003000200700000008004000100006102400080000070000090000
000010000020000030004506700003000800900000007007000500005703400010000060000020000

# 1-8-6-0-0
100050000006009000700030000200010503090600000000000040500020701030800000000000020
000010000020000030001405600003000400700000008006000500005603100080000070000090000
000010000020000030004506700003000500800000004007000600005703400010000090000020000

# 1-8-6-0-0
100050000006009000700030000200070305090600000000000040500020108030700000000000050
000010000020000030001405600003000200700000008004000100006201300080000070000090000
000010000020000030004506700003000500800000007007000900006703400010000060000020000

# 1-8-6-0-0
100050000006009000700030000200070305090600000000000040500020801030700000000000050
000010000020000030001405600003000200700000008004000100006102300080000070000090000
000010000020000030004506700003000800900000007007000500006703400010000060000020000

# 2-4-1-0-0
100050080000109000080000004000703000500010060030000002005000100007000900060000003
000102000001030400040000050300000001060000070800000002070000090003010600000807000
000102000003040500010000060200000006060000010300000007040000050008030900000607000

# 1-8-6-0-0
100050089006700000000000500030004000800010000900060000000000200500090016007300000
000010000020000030001405600003000400700000008006000500004306100080000020000090000
000010000020000030004506700003000500800000004007000600005307400010000020000090000

# 1-8-6-0-0
100050089006700000000000500030004000800060000900010000000000200500090016007300000
000010000020000030001405600003000400700000008006000500005603100080000020000090000
000010000020000030004506700003000500800000004007000600005703400010000020000090000

# 2-4-1-0-0
100050600000000073080003000030008000004700000008500000000000048010006000900020700
000102000001030400040000050300000001060000070700000002080000090003010600000807000
000102000003040500010000060200000007050000020300000004080000090004030800000407000

# 0-4-2-1-0
100400600400100700089000000000000000095008020000600400000000000072005090000300200
000000000001203400020000050030060070000108000090070060060000090002401800000000000
000000000001203400040000050050060020000107000080050040060000080003701900000000000

# 1-8-6-0-0
100450009006000700000002000070000060300000001900000004004000900000003000800510003
000010000020000030001405600003000200700000008004000100002503400080000070000090000
000010000020000030003405600004000700800000006007000500005306900010000040000020000

# 4-9-0-2-0
103000000406000000000200500000070090000004001030500700000090040000001002090600300
010000020300000004002050600000305000006000700000408000005010800700000003040000010
010000020300000004004050600000105000006000700000208000005030800200000003070000010

# 0-4-2-1-0
103050080000009700000000000000000000000004200608020010301000000040002900070006400
000000000001203400020000050030060070000108000090070060070000090002401800000000000
000000000001203400040000050050060020000701000080050040060000080003107900000000000

# 1-8-6-0-0
103050600000009070080000000000008030090000000607030200000200006000010500000070100
000010000020000030001405600003000200700000008004000100005204300080000070000060000
000010000020000030004506700003000500800000007005000400006703800010000060000020000

# 1-4-4-2-0
120000089400000030000600000000500000310000046500000010000084000002000700000021000
120000034300000005000050000000106000007000800000308000000020000400000001690000083
120000034400000005000050000000304000002000600000705000000080000600000009340000017

# 1-8-6-0-0
120400009000080070009000000000030050004000000810200007000900002000700008000005300
000010000020000030001405600003000200700000008004000100002503400080000070000060000
000010000020000030003405600004000700800000006007000500005306800010000040000020000
gsf
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Location: NJ USA

Postby Ocean » Tue May 09, 2006 2:47 pm

Havard wrote:Great work! Why don't you compile them all into a file, and make it available as "Ocean's 1100" or something... (sorry, could not resist...)

Havard
Thanks. The posted list is about maximum that can be published by posting here. Not sure how to make the rest of the puzzles available (if anybody should be interested), since I don't have a personal web-site.

JPF wrote:Huge work !
Congratulations.

I’ve a few questions :
a) Any new patterns ?
b) I assume that all your puzzles are non-equivalent. Right ?
I would be interested to know how you test that.
c) Would it be possible for some patterns to make an exhaustive search (notably to know the number of possible puzzles) ?

JPF

Glad you appreciate the collection! Hope it can be useful, either for analysis, or as a source for solvers.

a) No new patterns, the search was done in basically 'known' patterns, in order to produce a large collection.

b) The puzzles were supposed to be non-equivalent. I checked by normalizing them ("pattern normalization"), doing this by comparing the 2^4=16 equivalent arrangements that all patterns have (swap rows 4-6, swap columns 4-6, switch rows and columns, and turn left to right). Digits were assigned in specific order, first 1, then 2, etc. Among the 16 arrangements the 'lowest' one was selected as representing the puzzle. In retrospect I see that some patterns have extra symmetry, which I missed - and which may result in duplicates (doublets). There could also be other bugs in the newly written normalization procedure (wrote it yesterday).

c) Don't know how long time an exhaustive searh would take. Maybe the pattern symmetry could help optimize the search, but know how much.
Havard wrote:I ran my solver through most of these (1260), and found among other interesting stuff that these puzzles has an abnormal amount of single-digit eliminations! (like strong links, fishes, coloring etc)

I counted 5607 Box-Line / Line-Box eliminations, and 6548 eliminations of type x-wing, swordfish, 2/3/4 strong links, finned and franken variations etc. For 1260 puzzles that means quite a few of these per puzzle. I guess the extreme symmetry of these puzzle might be a possible cause of this?

Havard


Thanks for interesting statistics. I also observed that each pattern seems to have its own characteristic solving statistics.

gsf wrote:there are 65 isomorphic dups in ocean's post
the first puzzle is the canonical form used
(min row order lexicographic with top left box 123/456/789)
followed by the equivalent puzzles
in this case all dups were doubles


Thanks for checking isomorphism! As mentioned above, I missed one of the symmetries in some patterns (during normalization based on the pattern). Will look into it and eventually edit the posts (remove doubles).
Ocean
 
Posts: 442
Joined: 29 August 2005

Postby Ocean » Thu May 11, 2006 11:14 am

I have prepared a list of about 10000 non-isomorphic minimal 20s with full symmetry. Havard has been kind to offer to host the list, which is available here: Ocean20.txt. Eleven of ab's puzzles are included in the list, while the rest is my own compilation.
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Postby gsf » Thu May 11, 2006 2:52 pm

Ocean wrote:I have prepared a list of about 10000 non-isomorphic minimal 20s with full symmetry.

thanks
I doubled checked the isomorphisms -- none
only 6 pairs have isomorphic solution grids
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Postby Ocean » Thu May 11, 2006 4:06 pm

gsf wrote:thanks
I doubled checked the isomorphisms -- none
only 6 pairs have isomorphic solution grids

Thanks for the independent check. Did you notice whether the six pairs with isomorphic solution grids had the same or different patterns?
JPF wrote:b) I assume that all your puzzles are non-equivalent. Right ?
I would be interested to know how you test that.

Here is an update on the method I used: A fully symmetric puzzle can be transformed into 192 * 9! isomorphic forms where full symmetry is preserved. (For some patterns even more forms are possible, but that is not the case for this set.) Therefore: If we for all the 192 configurations 'normalize' the digits (by selecting 1 before 2 etc from start to end) we can pick a unique representation for that puzzle. This was done for all puzzles in the list. After this the task of excluding isomorphs becomes the simpler task: exclude duplicates in a sorted list.
#
Since gsf independently verified that all puzzles now are non-isomorphic, I am pretty confident that the simple procedure outlined above works, if used correctly.
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Postby gsf » Thu May 11, 2006 4:56 pm

Ocean wrote:Thanks for the independent check. Did you notice whether the six pairs with isomorphic solution grids had the same or different patterns?

here are the pairs with isomorphic solution grids
Code: Select all
"0-4-2-1-0   17" 000000000001000200020103040005060700000204000006050800010307090008000500000000000
"0-4-2-1-0   23" 000000000001000200020103040005060700000204000008070600090501030007000800000000000

"0-4-2-1-0  573" 000000000001000200030405060002010400000703000008060100050907030006000800000000000
"0-5-2-0-0 2549" 000000000001020300030405060006000700080000040007000100040806050009070200000000000

"0-4-2-1-0   22" 000000000001000200020103040005060700000204000008070600090501030006000800000000000
"0-4-2-1-0   18" 000000000001000200020103040005060700000204000006050800010307090008000600000000000

"0-5-2-0-0 3668" 000000000001020300040105060003000700060000080009000400050806010002030900000000000
"0-4-2-1-0   95" 000000000001000200020304050002060700000408000006070900030105040009000600000000000

"0-4-2-1-0   24" 000000000001000200020304010005060700000102000006050800040901020007000500000000000
"0-4-2-1-0   40" 000000000001000200020304050002010600000403000007060800070509030006000100000000000

"0-4-2-1-0   25" 000000000001000200020304010005060700000102000006050800040901020007000600000000000
"0-4-2-1-0   41" 000000000001000200020304050002010600000403000007060800070509040006000100000000000
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Here is my sudoku with a lean axis and a nice design

Postby claudiarabia » Sun May 14, 2006 11:16 am

Code: Select all
. 3 8 9 . . 4 . .
9 . . . 4 . 5 . .
4 . . . 6 . . 9 3
8 . . . 3 . . . .
. 4 3 5 . 9 2 6 .
. . . . 2 . . . 5
7 8 . . 9 . . . 4
. . 6 . 7 . . . 2
. . 4 . . 6 7 5 .
happy solving

Claudia
Last edited by claudiarabia on Tue Aug 08, 2006 3:34 pm, edited 4 times in total.
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Re: Here is my sudoku with a lean axis and a nice design

Postby JPF » Sun May 14, 2006 11:57 am

claudiarabia wrote:sorry it looks some odd for I have some problems with the format.
happy solving
Claudia

Code: Select all

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


Nice pattern, but it doesn't have all the symmetries .
So, it's not a F.S. puzzle.
Not abs. minimal.

See here for a description of the symmetries in Sudoku puzzles made by gfroyle.

JPF

PS : I like this type of pattern though:)
One more...
Code: Select all
 . 9 8 | 3 . . | . . .
 5 . . | . 6 . | . . .
 2 . . | . 7 . | . . .
-------+-------+-------
 7 . . | . . . | . . .
 . 3 1 | 6 . 8 | 2 9 .
 . . . | . . . | . . 4
-------+-------+-------
 . . . | . 1 . | . . 6
 . . . | . 4 . | . . 5
 . . . | . . 5 | 7 4 .
JPF
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Postby JPF » Wed May 24, 2006 2:04 pm

Ocean wrote:I have prepared a list of about 10000 non-isomorphic minimal 20s with full symmetry. Havard has been kind to offer to host the list, which is available here: Ocean20.txt. Eleven of ab's puzzles are included in the list, while the rest is my own compilation.

Well, it seems that your list has created some enthusiasm on the web:)

Ocean wrote:
JPF wrote:b) I assume that all your puzzles are non-equivalent. Right ?
I would be interested to know how you test that.

Here is an update on the method I used: A fully symmetric puzzle can be transformed into 192 * 9! isomorphic forms where full symmetry is preserved. (For some patterns even more forms are possible, but that is not the case for this set.) Therefore: If we for all the 192 configurations 'normalize' the digits (by selecting 1 before 2 etc from start to end) we can pick a unique representation for that puzzle...


As I always think of a question when it’s too late, I was wondering where 192 was coming from ?

Other question, if I may :
To generate puzzles based on a pattern, are you using a systematic type of search or a random process ?

As far as I’m concerned, I will shortly update the list of the valid patterns already found (20 to 25 clues) and hopefully add some more.

JPF
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Postby Ocean » Wed May 24, 2006 9:55 pm

JPF wrote:Well, it seems that your list has created some enthusiasm on the web:)
(...)
As far as I’m concerned, I will shortly update the list of the valid patterns already found (20 to 25 clues) and hopefully add some more.
Good. By the way, I now have a collection of more than 15000 fully symmetrical puzzles with 20 clues. But it's probably too early for an update. Regarding the 'enthusiasm on the web', I have not noticed any references outside this forum.

JPF wrote:
Ocean wrote:
JPF wrote:b) I assume that all your puzzles are non-equivalent. Right ?
I would be interested to know how you test that.

Here is an update on the method I used: A fully symmetric puzzle can be transformed into 192 * 9! isomorphic forms where full symmetry is preserved. (For some patterns even more forms are possible, but that is not the case for this set.) Therefore: If we for all the 192 configurations 'normalize' the digits (by selecting 1 before 2 etc from start to end) we can pick a unique representation for that puzzle...

As I always think of a question when it’s too late, I was wondering where 192 was coming from ?

An attempt (reasoning, but no proof) ... :

Full symmetry is always preserved when (any combination of) this subset of the equivalence transformations is performed:
1. Simultaneous permutation of rows 1-3, rows 7-9, columns 1-3, columns 7-9. (Factor 6).
2. Turn the board upside down. (Factor 2).
3. Turn the board left-to-right. (Factor 2).
4. Switch rows 4 and 6. (Factor 2).
5. Switch columns 4 and 6. (Factor 2).
6. Turn the board 90 degrees (rows become columns). (Factor 2).

The factor 192 is the combination of all these (6*2^5=192).

It should be noted that some specific patterns preserve full symmetry with other transformations as well. Therefore care has to be taken before concluding that checking these 192 variations guarantees isomorphism. It's not valid for all patterns.

Other transformations than those listed may be chosen as a basis, but they can be expressed a combination of those listed (and vice versa). The first transformation listed (#1) normally does not preserve the pattern, but serves as a normalization, catching 'equivalent patterns'.

JPF wrote:Other question, if I may :
To generate puzzles based on a pattern, are you using a systematic type of search or a random process ?
As I have understood, this is not the appropriate forum for discussing generating algorithms. I have experimented with several algorithms - sometimes they give results, sometimes not.
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Postby ab » Wed Jun 21, 2006 2:02 am

A minimal version of 0-2-5-2-1
Code: Select all
 . . . | . . . | . . .
 . . . | 8 . 6 | . . .
 . . 3 | . 2 . | 5 . .
 ------+-------+------
 . 8 . | 2 . 9 | . 1 .
 . . 7 | . 4 . | 3 . .
 . 9 . | 1 . 3 | . 6 .
 ------+-------+------
 . . 4 | . 5 . | 2 . .
 . . . | 9 . 8 | . . .
 . . . | . . . | . . .

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Postby JPF » Wed Jun 21, 2006 2:43 pm

Nice improvement.

I updated the list.
Some 20 & 21 clues patterns are still available...

JPF
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Postby tso » Wed Jun 21, 2006 2:48 pm

Although puzzles like this have only diagonal symmetry, the *clue pattern* is fully symmetrical:

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


That ought'a be worth sumpin'.
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Postby Ocean » Mon Jul 17, 2006 9:35 am

Collections of fully symmetric puzzles with 20 clues can be a handy source for finding puzzles with specific properties. From easy/medium to advanced and very hard. Here is some statistics.

1. Easy to Medium.

There are plenty of easy/medium puzzles in this colleciton, from low-steppers to highsteppers.

Count of single-steppers in the 10117-collection:
Code: Select all
 
-----------------------------------------------------------------------------
| 8|  9| 10| 11| 12| 13| 14| 15| 16| 17| 18| 19| 20| 21| 22| 23| 24| 25| 26|  # steps
-----------------------------------------------------------------------------
| 2| 42|164|410|576|681|563|433|354|258|162| 80| 43| 25|  9|  2|  1|  1|  2|  # puzzles
-----------------------------------------------------------------------------


2. Medium to Advanced.

Many puzzles are found in this range. For instance this puzzle by ab from the Superior Thread.

Some counting/statistics (the 10117-collection):

Puzzles that solve with locked candidates and/or naked pairs: 1573
Puzzles that solve with X-Wing as 'highest technique': 82
Puzzles that solve with XY-Wing as 'highest technique': 534
Puzzles with small XY-ring (4 nodes): 46
Puzzles with short XY-chain (cycle-lenght 5): 290

I do not have a counter for Swordfish or other specialized techniques so would appreciate if somebody could provide stats.

3. Very hard puzzles.
One of the patterns contains some very hard puzzles, even aspiring for the hardest knowns. In case someone should want to study such puzzles: Here is a list of 50 puzzles which Sudoku Explainer rates 9.0 or higher, all in the same pattern.

001000200020030010400000005000406000070000080000905000300000004080010070002000900 #ER=9.0
001000200020030010400000005000406000070000080000905000500000009080010070003000600 #ER=9.2
001000200020030040500000006000506000010000070000809000600000008070020010003000900 #ER=9.1
001000200020030040500000006000506000010000070000809000800000005070040010004000300 #ER=9.0 001000200020030040500000006000506000010000070000809000900000008040020010003000500 #ER=9.0
001000200020030040500000006000507000010000080000906000600000009080020010003000700 #ER=9.1
001000200020030040500000006000507000010000080000609000600000009080020010004000700 #ER=9.1
001000200020030040500000006000507000010000080000609000900000007040020010003000500 #ER=9.0
001000200020030040500000006000507000010000080000906000900000005040020010003000700 #ER=9.0
001000200020030040500000006000507000010000080000906000900000007040020010003000500 #ER=9.0
001000200030010040500000006000507000040000080000609000700000009080030010002000500 #ER=9.1
001000200030040050600000007000103000020000080000705000700000006080050040009000100 #ER=9.0
001000200030040050600000007000103000040000030000604000700000001080050040002000600 #ER=9.0
001000200030040050600000007000103000050000010000602000700000006040080030002000900 #ER=9.0
001000200030040050600000007000103000050000030000208000200000001090030040007000800 #ER=9.0
001000200030040050600000007000103000050000030000602000200000001080030040007000900 #ER=9.0
001000200030040050600000007000103000050000030000602000700000006080030040002000900 #ER=9.0
001000200030040050600000007000103000050000030000608000700000001090030040002000600 #ER=9.0
001000200030040050600000007000103000050000040000604000700000006040080030002000100 #ER=9.2
001000200030040050600000007000103000050000040000704000700000008040090030002000100 #ER=9.1
001000200030040050600000007000103000050000080000507000700000006040050030002000100 #ER=9.3
001000200030040050600000007000103000050000080000604000200000001080090040007000600 #ER=9.1
001000200030040050600000007000103000050000080000604000700000001040090030002000600 #ER=9.1
001000200030040050600000007000103000050000080000605000700000006040050030002000100 #ER=9.2
001000200030040050600000007000103000050000080000609000700000006080030040002000100 #ER=9.1
001000200030040050600000007000103000050000080000705000700000009080030040002000100 #ER=9.2
001000200030040050600000007000103000050000080000907000700000001040020030008000900 #ER=9.0
001000200030040050600000007000103000080000030000506000700000001050090080002000600 #ER=9.3
001000200030040050600000007000103000080000030000506000700000006050030080009000100 #ER=9.7
001000200030040050600000007000103000080000030000602000700000001050030040002000900 #ER=9.0
001000200030040050600000007000103000080000030000602000700000001050030080002000900 #ER=9.0
001000200030040050600000007000103000080000030000602000900000001040050080007000600 #ER=9.0
001000200030040050600000007000103000080000030000604000200000006040050080007000100 #ER=9.8
001000200030040050600000007000103000080000030000608000700000006050030040002000100 #ER=9.4
001000200030040050600000007000103000080000030000609000700000006040050080009000100 #ER=9.2
001000200030040050600000007000103000080000030000709000700000006040050080009000100 #ER=9.0
001000200030040050600000007000103000080000090000507000700000001090050040002000800 #ER=9.0
001000200030040050600000007000103000080000090000605000200000006050030040007000100 #ER=9.1
001000200030040050600000007000104000040000030000602000200000008090050040007000600 #ER=9.0
001000200030040050600000007000104000040000030000605000700000001050080090002000600 #ER=9.2
001000200030040050600000007000104000040000080000605000700000001050030090002000600 #ER=9.2
001000200030040050600000007000104000050000040000608000700000006040030090002000100 #ER=9.0
001000200030040050600000007000104000050000040000608000700000006080090030002000100 #ER=9.3 001000200030040050600000007000104000050000040000703000700000008040090030002000100 #ER=9.1
001000200030040050600000007000104000050000080000709000700000006040030090008000100 #ER=9.0
001000200030040050600000007000104000080000030000708000700000006040050080009000100 #ER=9.2
001000200030040050600000007000104000080000030000907000200000009050060080007000100 #ER=9.0
001000200030040050600000007000104000080000090000605000700000006050030080002000100 #ER=9.1
001000200030040050600000007000105000040000080000904000700000006050030040002000100 #ER=9.6
001000200030040050600000007000105000080000090000608000200000001090030040007000600 #ER=9.1
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