## Fully symmetrical puzzles

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

Posts: 377
Joined: 25 December 2005

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
`# 1-4-4-2-0000000009450109000080006000070005000530704000000000008000000490000000530002070000120000034300000002000050000000102000006000300000704000000070000500000008240000097120000034500000006000010000000103000007000800000904000000060000400000007370000049# 2-4-1-0-0000000680050100007009003000090700003000000420008002000600040000800070000002008000000102000001030400040000050300000001060000070800000002070000090003010600000708000000102000003040500010000060200000007050000010300000004080000090004030800000407000# 4-9-0-2-0000007600000100020009030004000300070005040002000006100360000000570000000000020008010000020300000004004050600000102000007000800000804000006090500500000001020000030010000020300000004004050600000203000007000800000907000006040900900000001020000030# 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1-8-6-0-0020007000400100000080006000001040000000000900070002038000000800007020000030008056000010000020000030001405600003000200700000008004000100002106300080000070000090000000010000020000030003405600006000400700000006008000900005603800010000050000020000# 1-8-6-0-0020007000400100000080006000001040000000000900070002038000000800007020000030008065000010000020000030001405600003000200700000008004000100002601300080000070000090000000010000020000030003405600006000700800000006009000400005603900010000050000020000# 4-9-0-2-0020007080400080000009300000000000403000000802070001000300040000008900000010002050010000020300000004005010600000106000007000500000203000006080700400000001020000030010000020300000004005030600000104000007000500000603000006080700400000001020000030# 4-9-0-2-0020007600400080000009300000008900000300040000060002900000000028030005000000000014010000020300000004005010600000203000004000500000706000006080900400000001020000030010000020300000004005030600000104000002000500000607000006080900400000001020000030# 4-9-0-2-0020400000000000037000000015005010000070900800600003000040800900800005000001060000010000020300000004002050600000104000006000700000809000005020800400000001090000030010000020300000004004050600000203000006000700000809000005040900800000001020000030# 4-9-0-2-0020400000000000073000000051001060000070900800900005000005010000600003000040700300010000020300000004004050600000203000007000800000906000006090700900000001020000030010000020300000004005030600000102000006000700000305000008090500200000008040000010# 4-9-0-2-0020400009000080070000003500000600007601000000805000000000005800040700001000010030010000020300000004002050600000207000006000700000403000008090500400000001060000030010000020300000004004050600000102000006000700000704000008090500600000001020000030# 4-9-0-2-0020400009006080000700002000000000750040300000000000820300006000008070000090500001010000020300000004002050600000107000006000800000402000005090700400000001080000030010000020300000004004050600000204000006000700000308000005090800700000001020000030# 0-4-2-1-0020450009006000030000000000007001060000240000008005010070890004000000000001000070000000000001203400020000050030050060000107000080060070060000080002401900000000000000000000001203400040000050050060020000701000080050040090000080003106700000000000# 1-4-4-2-0023000000000100070089000000200000000000003400000045308000008900600000000000031504120000034300000001000050000000105000006000700000802000000090000800000002430000017120000034300000005000010000000206000007000600000803000000090000500000002480000053# 1-4-4-2-0023406000006700000000000500000000067000000021900010000071608000000000300004200000120000034300000005000060000000207000008000700000305000000010000400000002750000093120000034400000005000060000000206000007000800000403000000090000300000002840000016# 4-9-0-2-0100000009006700020080000500200000001090000800007600030000014000000093000002000050010000020300000004005010600000203000006000500000708000008090100400000009020000030010000020300000004005030600000104000006000500000708000007090300400000001020000090# 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2-4-1-0-0100050080000109000080000004000703000500010060030000002005000100007000900060000003000102000001030400040000050300000001060000070800000002070000090003010600000807000000102000003040500010000060200000006060000010300000007040000050008030900000607000# 1-8-6-0-0100050089006700000000000500030004000800010000900060000000000200500090016007300000000010000020000030001405600003000400700000008006000500004306100080000020000090000000010000020000030004506700003000500800000004007000600005307400010000020000090000# 1-8-6-0-0100050089006700000000000500030004000800060000900010000000000200500090016007300000000010000020000030001405600003000400700000008006000500005603100080000020000090000000010000020000030004506700003000500800000004007000600005703400010000020000090000# 2-4-1-0-0100050600000000073080003000030008000004700000008500000000000048010006000900020700000102000001030400040000050300000001060000070700000002080000090003010600000807000000102000003040500010000060200000007050000020300000004080000090004030800000407000# 0-4-2-1-0100400600400100700089000000000000000095008020000600400000000000072005090000300200000000000001203400020000050030060070000108000090070060060000090002401800000000000000000000001203400040000050050060020000107000080050040060000080003701900000000000# 1-8-6-0-0100450009006000700000002000070000060300000001900000004004000900000003000800510003000010000020000030001405600003000200700000008004000100002503400080000070000090000000010000020000030003405600004000700800000006007000500005306900010000040000020000# 4-9-0-2-0103000000406000000000200500000070090000004001030500700000090040000001002090600300010000020300000004002050600000305000006000700000408000005010800700000003040000010010000020300000004004050600000105000006000700000208000005030800200000003070000010# 0-4-2-1-0103050080000009700000000000000000000000004200608020010301000000040002900070006400000000000001203400020000050030060070000108000090070060070000090002401800000000000000000000001203400040000050050060020000701000080050040060000080003107900000000000# 1-8-6-0-0103050600000009070080000000000008030090000000607030200000200006000010500000070100000010000020000030001405600003000200700000008004000100005204300080000070000060000000010000020000030004506700003000500800000007005000400006703800010000060000020000# 1-4-4-2-0120000089400000030000600000000500000310000046500000010000084000002000700000021000120000034300000005000050000000106000007000800000308000000020000400000001690000083120000034400000005000050000000304000002000600000705000000080000600000009340000017# 1-8-6-0-0120400009000080070009000000000030050004000000810200007000900002000700008000005300000010000020000030001405600003000200700000008004000100002503400080000070000060000000010000020000030003405600004000700800000006007000500005306800010000040000020000`
gsf
2014 Supporter

Posts: 7306
Joined: 21 September 2005
Location: NJ USA

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

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.
Ocean

Posts: 442
Joined: 29 August 2005

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
gsf
2014 Supporter

Posts: 7306
Joined: 21 September 2005
Location: NJ USA

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.
Ocean

Posts: 442
Joined: 29 August 2005

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`
gsf
2014 Supporter

Posts: 7306
Joined: 21 September 2005
Location: NJ USA

### Here is my sudoku with a lean axis and a nice design

Code: Select all
`. 3 8 9 . . 4 . . 9 . . . 4 . 5 . . 4 . . . 6 . . 9 38 . . . 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.
claudiarabia

Posts: 288
Joined: 14 May 2006

### Re: Here is my sudoku with a lean axis and a nice design

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
2017 Supporter

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

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

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.
Ocean

Posts: 442
Joined: 29 August 2005

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 | . . .  . . . | . . . | . . . `
ab

Posts: 451
Joined: 06 September 2005

Nice improvement.

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

JPF
JPF
2017 Supporter

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

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'.
tso

Posts: 798
Joined: 22 June 2005

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----------------------------------------------------------------------------- `

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
Ocean

Posts: 442
Joined: 29 August 2005

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