## Fully symmetrical puzzles

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

### Fully symmetrical puzzles

Thanks Gordon for the description of symmetries in puzzles using the Group theory.

In some recent threads (or this one), we saw many fully symmetrical puzzles.
Like this one :
Code: Select all
` . . . | . 7 . | . . . . . 5 | . . . | 9 . . . 3 . | 5 . 4 | . 2 .-------+-------+------- . . 3 | . 4 . | 1 . . 4 . . | 3 . 1 | . . 9 . . 1 | . 5 . | 8 . .-------+-------+------- . 8 . | 7 . 2 | . 5 . . . 9 | . . . | 6 . . . . . | . 1 . | . . .`

with the pattern :
Code: Select all
` . . . . x . . . .  . . x . . . x . . . x . x . x . x . . . x . x . x . . x . . x . x . . x . . x . x . x . . . x . x . x . x . . . x . . . x . . . . . . x . . . . `
Lots of them are very nice.

As far as I'm concern, I'm happy with some of my collection...
Here is one :

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

or this one :

Code: Select all
` 6 . . | 3 1 9 | . . 7  . . . | 2 . 4 | . . .  . . . | . . . | . . . -------+-------+------- 1 8 . | . . . | . 6 9  7 . . | . 6 . | . . 1  2 4 . | . . . | . 7 5 -------+-------+------- . . . | . . . | . . .  . . . | 4 . 8 | . . .  9 . . | 7 5 3 | . . 6`
... even if tarek doesn't like them...

but stuck with those 2 patterns, unable to get a valid puzzle (each of those 2 examples has 2 solutions).
Code: Select all
` . . . | 2 8 5 | . . . . . 4 | . . . | 2 . . . 6 . | . . . | . 3 .-------+-------+------- 5 . . | . . . | . . 4 3 . . | . 7 . | . . 8 2 . . | . . . | . . 9-------+-------+------- . 1 . | . . . | . 7 . . . 9 | . . . | 5 . . . . . | 3 1 2 | . . .`

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

I was wondering how many fully symmetrical patterns have at least one valid puzzle. In that case, I would call it a "fully symmetrical valid pattern"(FSVP).
More precisely, the idea would be to set up a list of all them (where ?), with an example for each one (a minimal puzzle, if possible).

So, my first move was to count the number of fully symmetrical patterns.

It's 2^15=32768, as Gordon recall it, ( the whole dihedral group D_8).

More precisely, let N(c) be the number of patterns with c clues (0<=c<=81).
We have the following relations :

N(81-c)=N(c) ; c = 0,...,81 (1)
N(4k)=N(4k+1) ; k = 0,...,20 (2)
N(4k+2)=N(4k+3)=0 ; k = 0,...,19 (3)

And the distribution of N(c) can be summarized in the table below :

Code: Select all
`Number            Numberof clue         of patterns(c)               N(c)0                   11                   14                   85                   88                  349                  3412                10413                10416                25317                25320                51221                51224                88825                88828               134429               134432               179433               179436               212837               212840               225241               2252`

The remaining values can be calculated by symmetry (!), using the relation (1).

Evidently, each pattern with 4k+1 clues is made of one pattern with 4k clues and an x in R5C5. (2)

Assuming (conjecture A) that the minimum clues number for a valid puzzle is 17 and that there is no symmetrical puzzle with 17 clues (conjecture B), there is no FSVP for n<20.
Assuming (conjecture C) that the maximum clues for a minimal puzzle is... let's say 37 to be "safe" :

we need to look at 2x(512+888+1344+1794+2128)=2x6666=13332 patterns only !

It works fine if, for a given pattern, a valid puzzle is found quickly , and will be painfull if not.

JPF
JPF
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Location: Paris, France

### Re: Fully symmetrical puzzles

JPF wrote:I was wondering how many fully symmetrical patterns have at least one valid puzzle. In that case, I would call it a "fully symmetrical valid pattern"(FSVP).
More precisely, the idea would be to set up a list of all them (where ?), with an example for each one (a minimal puzzle, if possible).

JPF

I assume you will end up with a large catalog. Here is a modest start (three minimal puzzles with 24 clues):

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

Posts: 442
Joined: 29 August 2005

JPF wrote:we need to look at 2x(512+888+1344+1794+2128)=2x6666=13332 patterns only !

Have you considered that simultaneous permutation of rows 1-3 & 7-9 and columns 1-3 & 7-9 allow you to reduce the number of patterns that you need to find?

Ruud.
Ruud

Posts: 664
Joined: 28 October 2005

### Re: Fully symmetrical puzzles

JPF wrote:even if tarek doesn't like them...

I liked them, however they were too tough for the thread however...
this modified one....
Code: Select all
`JPF/tarek 6 . . | 3 1 9 | . . 7   . 1 . | 2 . 4 | . 5 .   . . . | . . . | . . .  -------+-------+------  1 8 . | . . . | . 6 9   7 . . | . 2 . | . . 1   2 4 . | . . . | . 7 5  -------+-------+------  . . . | . . . | . . .   . 6 . | 1 . 8 | . 3 .   9 . . | 7 5 3 | . . 6`
keeps the full symmetry, & is easier & will go on the list with another modification that will lose the full symmetry but keeping the 180 rotational symmetry

tarek

tarek

Posts: 3761
Joined: 05 January 2006

### Re: Fully symmetrical puzzles

JPF wrote:I was wondering how many fully symmetrical patterns have at least one valid puzzle. In that case, I would call it a "fully symmetrical valid pattern"(FSVP).
More precisely, the idea would be to set up a list of all them (where ?), with an example for each one (a minimal puzzle, if possible).

Two fully symmetrical patterns with 25 clues, illustrated with minimal 25s.
Code: Select all
` . 5 . | . . . | . 9 . 6 8 . | . . . | . 4 2  . . . | 7 . 2 | . . . -------+-------+------  . . 6 | . 3 . | 5 . .  . . . | 2 1 4 | . . . . . 8 | . 9 . | 4 . .-------+-------+------  . . . | 9 . 3 | . . .  2 1 . | . . . | . 5 8 . 4 . | . . . | . 6 . ###################### . . 4 | . . . | 7 . .  . 1 . | . 4 . | . 6 . 6 . 7 | . . . | 1 . 9 -------+-------+------  . . . | 9 . 2 | . . .  . 6 . | . 3 . | . 9 .  . . . | 8 . 6 | . . .-------+-------+------  1 . 6 | . . . | 8 . 5  . 5 . | . 2 . | . 7 . . . 3 | . . . | 2 . . `

Also two patterns with 28 clues, illustrated with minimal 28s.
Code: Select all
` 3 . 2 | . . . | 1 . 5  . . 4 | . . . | 8 . .  5 7 . | . . . | . 2 9 -------+-------+------  . . . | 5 1 7 | . . .  . . . | 8 . 9 | . . .  . . . | 3 2 6 | . . . -------+-------+------  4 9 . | . . . | . 3 2  . . 5 | . . . | 9 . .  2 . 8 | . . . | 7 . 1 ###################### 8 . . | . 6 . | . . 9  . 9 . | 1 . 7 | . 8 .  . . . | 3 . 8 | . . . -------+-------+------  . 4 8 | . . . | 3 5 .  5 . . | . . . | . . 4  . 1 2 | . . . | 6 7 . -------+-------+------  . . . | 6 . 4 | . . . . 5 . | 8 . 2 | . 4 .  3 . . | . 5 . | . . 7 `

And one pattern with 29 clues, illustrated with a minimal 29.
Code: Select all
` . . . | . . . | . . . . 6 . | 1 . 4 | . 7 .  . . 9 | 7 . 8 | 4 . .-------+-------+------  . 9 2 | 4 . 6 | 5 1 .  . . . | . 5 . | . . . . 7 5 | 9 . 1 | 6 2 .-------+-------+------  . . 6 | 3 . 9 | 2 . . . 8 . | 2 . 7 | . 3 . . . . | . . . | . . .`
Ocean

Posts: 442
Joined: 29 August 2005

A bit busy these days...

Ocean wrote:Here is a modest start (three minimal puzzles with 24 clues)
...
Two fully symmetrical patterns with 25 clues, illustrated with minimal 25s.
...
Also two patterns with 28 clues, illustrated with minimal 28s.
...
And one pattern with 29 clues, illustrated with a minimal 29.

Many thanx for your 8 FS minimal puzzles.
They are really brilliant !
Ruud wrote:Have you considered that simultaneous permutation of rows 1-3 & 7-9 and columns 1-3 & 7-9 allow you to reduce the number of patterns that you need to find?

No, I haven't, but you are right. Thanks.
These permutations reduce the number of patterns to be considered.

In order to list the FS patterns, I'm using the following rule :
One pattern is entirely defined by 15 bits (ie, blank (0) or x (1) in RiCj with 1<=i<=j=5).
Let t(i,j) be the bit in RiCj.( 1<=i<=j=5)
The pattern can be characterized by 5 numbers a,b,c,d,e :

Code: Select all
`a=t(1,1)t(1,2)t(1,3)t(1,4)t(1,5)        0<=a<=31b=t(2,2)t(2,3)t(2,4)t(2,5)              0<=b<=15c.......................................0<=c<=7d.......................................0<=d<=3e=t(5,5)                                0<=e<=1`

So, for instance, this Ocean's puzzle
Code: Select all
` . . . | 1 2 3 | . . . . . 3 | . . . | 4 . . . 2 . | . . . | . 5 .-------+-------+------- 6 . . | . 3 . | . . 5 7 . . | 8 . 9 | . . 2 1 . . | . 7 . | . . 9-------+-------+------- . 9 . | . . . | . 8 . . . 5 | . . . | 2 . . . . . | 2 4 6 | . . .`

Code: Select all
`Row 1 :00011 = 3Row 2 : 0100 = 4Row 3 :..000 = 0Row 4 :...01 = 1Row 5 : ...0 = 0`

and the pattern : 3-4-0-1-0
the puzzle itself :
Code: Select all
`000123000003000400020000050600030005700809002100070009090000080005000200000246000`
We can add 24 clues, M (for minimal)

Circles.
In a first step, one can consider the patterns delimited by one circle of radius equal to 4, like Ocean's one mentioned above.
These patterns are defined by the following code :

3-b-c-d-e
4<=b<=7; 0<=c<=7; 0<=d<=3; 0<=e<=1

There are 2^8 = 256 possible patterns.
The distribution of the number of clues is :
Code: Select all
`Number      Numberof clues      of patterns20            121            124            525            528            1229            1232            2033            2036            2637            2640            2641            2644            2045            2048            1249            1252            553            556            157            1`

Here are some examples :

3- 4- 0- 2- 0 ; 24 clues, NSM (not symetrically minimal) :
Code: Select all
` . . . | 2 1 9 | . . . . . 6 | . . . | 2 . . . 4 . | . . . | . 9 .-------+-------+------- 9 . . | 1 . 8 | . . 5 5 . . | . . . | . . 6 6 . . | 7 . 2 | . . 3-------+-------+------- . 1 . | . . . | . 6 . . . 2 | . . . | 4 . . . . . | 9 7 5 | . . .`
(The minimum one from Ocean above is a 6- 8- 0- 2- 0 ; 24 clues , M ; Not a real circle)

3- 4- 2- 0- 0 ; 28 clues, NSM :
Code: Select all
` . . . | 1 2 7 | . . . . . 4 | . . . | 7 . . . 3 . | 8 . 4 | . 2 .-------+-------+------- 3 . 7 | . . . | 6 . 9 5 . . | . . . | . . 3 4 . 1 | . . . | 2 . 7-------+-------+------- . 2 . | 7 . 8 | . 3 . . . 5 | . . . | 9 . . . . . | 5 1 6 | . . .`

3- 4- 3- 0- 0 ; 32 clues, NSM :
Code: Select all
` . . . | 6 9 8 | . . . . . 2 | . . . | 9 . . . 9 . | 2 5 3 | . 4 .-------+-------+------- 7 . 5 | . . . | 4 . 3 2 . 6 | . . . | 5 . 7 3 . 9 | . . . | 1 . 8-------+-------+------- . 7 . | 5 6 4 | . 1 . . . 3 | . . . | 6 . . . . . | 3 7 2 | . . .`

3- 5- 2- 2- 0 ; 36 clues, NSM :
Code: Select all
` . . . | 5 4 2 | . . . . . 1 | . 3 . | 2 . . . 6 . | 1 . 8 | . 9 .-------+-------+------- 2 . 6 | 4 . 1 | 7 . 9 7 1 . | . . . | . 8 2 8 . 5 | 2 . 7 | 3 . 4-------+-------+------- . 4 . | 8 . 9 | . 2 . . . 8 | . 1 . | 5 . . . . . | 6 2 4 | . . .`

3- 7- 7- 0- 0 ; 48 clues, NSM :
Code: Select all
` . . . | 9 6 7 | . . . . . 6 | 1 2 3 | 9 . . . 9 7 | 4 8 5 | 3 6 .-------+-------+------- 3 2 9 | . . . | 1 7 5 8 6 5 | . . . | 4 9 3 1 7 4 | . . . | 2 8 6-------+-------+------- . 5 8 | 2 3 4 | 7 1 . . . 3 | 6 9 1 | 8 . . . . . | 5 7 8 | . . .`

until the 3- 7- 7- 3- 1 ; 57s clues :
Code: Select all
` . . . | 7 2 8 | . . . . . 7 | 4 3 1 | 5 . . . 3 8 | 6 9 5 | 2 7 .-------+-------+------- 7 5 3 | 8 1 4 | 6 9 2 9 8 1 | 5 6 2 | 4 3 7 6 4 2 | 3 7 9 | 8 5 1-------+-------+------- . 7 9 | 2 8 6 | 1 4 . . . 4 | 9 5 7 | 3 . . . . . | 1 4 3 | . . .`

Unfortunately, all my puzzles are not SM, but very easy to solve

Code: Select all
` . . . | x x x | . . . . . x | . . . | x . . . x . | . . . | . x .-------+-------+------- x . . | . . . | . . x x . . | . x . | . . x x . . | . . . | . . x-------+-------+------- . x . | . . . | . x . . . x | . . . | x . . . . . | x x x | . . .`

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

### Re: Fully symmetrical puzzles

Sudokus with fullsymetric patterns with 20 clues are quite rare, but here is one example.

Code: Select all
`20 clues. Minimal. Full symmetry. . . . | . . . | . . .  . . 1 | . 2 . | 3 . .  . 4 . | 5 . 6 | . 7 . -------+-------+------  . . 4 | . . . | 1 . .  . 7 . | . . . | . 8 .  . . 2 | . . . | 6 . . -------+-------+------  . 8 . | 9 . 7 | . 5 .  . . 3 | . 6 . | 2 . .  . . . | . . . | . . . `
Ocean

Posts: 442
Joined: 29 August 2005

Ocean wrote:Sudokus with fullsymetric patterns with 20 clues are quite rare, but here is one example.

Wonderful !
0-5-2-0-0 ; 20 clues ; M (minimal)
You produce so much splendid grids that I have a great deal of difficulty…
… to classify them.

Ok, let’s try the 20-clues fully symmetrical patterns.
But, unfortunately, I have no contribution at the moment

Here is one posted by ab in an other thread :
Code: Select all
` 7 9 . | . . . | . 5 4 5 . . | . . . | . . 7 . . . | . 1 . | . . .-------+-------+------- . . . | 1 . 2 | . . . . . 8 | . . . | 1 . . . . . | 5 . 7 | . . .-------+-------+------- . . . | . 9 . | . . . 6 . . | . . . | . . 3 2 4 . | . . . | . 7 9`

24- 0- 1- 2- 0 ; 20 clues ; SM (symmetric minimal)
Is it possible to make it absolutely minimal ?

JPF
JPF
2017 Supporter

Posts: 6001
Joined: 06 December 2005
Location: Paris, France

here is a list of minimal full symmetric puzzles generated overnight
sorted and uniq'd by your nomenclature
a neat 28 clue minimal popped out
Code: Select all
`2 7 .  . . .  . 9 16 . 8  . . .  5 . 7. 9 .  . . .  . 6 .. . .  3 1 5  . . .. . .  2 . 9  . . .. . .  6 4 8  . . .. 3 .  . . .  . 7 .4 . 6  . . .  9 . 85 1 .  . . .  . 3 4`

and the list

000000000004030500030106080003709400010000050007804300080201030002070900000000000 # 0-5-2-2-0
000000000008702600010805070037000940000040000025000130050407020001308500000000000 # 0-6-2-0-1
000000000008605900027000350040090060000402000010070080069000520005809400000000000 # 0-6-4-1-0
000000000007803500045000860070050090000346000060020080092000150004905700000000000 # 0-6-4-1-1
000000000001849200092000180020060010030408060080020030014000870008594600000000000 # 0-7-4-1-0
000000000031000960060504030007030600000605000002010700040209010086000340000000000 # 0-12-2-1-0
000000000045000780090308040002040900000819000001060300030401090014000250000000000 # 0-12-2-1-1
000000000039000450010806020006302100000050000003901800090105040042000730000000000 # 0-12-2-2-1
000060000005000900040903060002090300900804001007010200090501080006000700000040000 # 1-4-2-1-0
000090000008000500020408090007905600400000008001806400070601050003000100000030000 # 1-4-2-2-0
000080000002309500060000070090030020500402001070060080040000010003704600000090000 # 1-6-0-1-0
000020000003608700020403080058000670700000001061000820080307010007901500000040000 # 1-6-2-0-0
000708000000591000006000500970000026060000050310000078002000400000236000000107000 # 2-3-4-0-0
000506000001000200050080060800030007002608400900010003080050070007000900000301000 # 2-4-1-1-0
000104000006000400090060070500403007008000100900205004070020010003000500000609000 # 2-4-1-2-0
000105000007000500090408020402000305000000000308000204030209060006000100000503000 # 2-4-2-0-0
000407000001000200070103050102000804000090000409000603040501020006000100000809000 # 2-4-2-0-1
000407000006000100048000950500010006000208000900030002029000840001000700000503000 # 2-4-4-1-0
000307000001000900092000860500609002000000000900704001018000540004000300000502000 # 2-4-4-2-0
000603000001000300078000240200504009000090000900708002037000820006000400000401000 # 2-4-4-2-1
000209000002010500060000010200090006030105070100080004070000090005060800000302000 # 2-5-0-1-0
000201000001080500040000060200307009070000020900506001050000030009010200000403000 # 2-5-0-2-0
000903000002080600030000040100706009050040020900108003080000070005010200000205000 # 2-5-0-2-1
000705000006020100050090030800000007041000320200000001030060040005030200000902000 # 2-5-1-0-0
000304000004060300050010070400000009076090210100000008060050030005020100000709000 # 2-5-1-0-1
000107000004906300030000060910000026000000000850000017040000070007804500000609000 # 2-6-0-0-0
000801000008307200070000030960000057000060000540000061090000040005403700000106000 # 2-6-0-0-1
000907000060000040009080300700030002001509600300070009008060200040000090000103000 # 2-8-5-1-0
000604000050000080001308700407000306000060000905000801008109400070000090000705000 # 2-8-6-0-1
000609000020050040009000800600080007040506020200040005004000900030060010000702000 # 2-9-4-1-0
000209000080060070006000200300501008010000030600902004001000600090040080000807000 # 2-9-4-2-0
000807000060050070001020800600000003092000650700000004003010900020070010000603000 # 2-9-5-0-0
000503000060804020008000600740000052000020000920000046009000300050608010000701000 # 2-10-4-0-1
000107000051000280030000050900060005000209000100070006060000030048000190000402000 # 2-12-0-1-0
000204000073000690020000080800401005000000000900603001080000010061000430000905000 # 2-12-0-2-0
000304000012000950070000020900603008000040000100809004080000060065000340000206000 # 2-12-0-2-1
000906000035708920080000070870000045000000000260000089090000060048107350000805000 # 2-14-0-0-0
000132000002000800050000040700050003400809005500060008070000010003000600000625000 # 3-4-0-1-0
000235000009000800030000010800407002100000004400502006070000050001000600000986000 # 3-4-0-2-0
000914000002060800040000060700000002960050037300000001080000020003090100000725000 # 3-5-0-0-1
000216000070000080009000100900050002400601003300080007005000600010000040000967000 # 3-8-4-1-0
000964000070000090008000500200408005900000007700102008009000100030000040000786000 # 3-8-4-2-0
002000100000060000800507003003020700020803090001090800700208001000030000005000400 # 4-1-2-1-0
004000200000302000900080005080020060003106800070040010800010004000607000002000600 # 4-2-1-1-0
001000700000509000400080003010305020006000500070802010300010007000208000005000900 # 4-2-1-2-0
003000200000204000600903008059000130000000000086000450400809002000107000005000600 # 4-2-2-0-0
003000100000509000600807003058000340000050000021000680100603008000705000006000900 # 4-2-2-0-1
009000600000901000504000807030020080000107000080050040806000102000706000007000900 # 4-2-4-1-0
007000500000105000109000603010020030000798000090050020504000801000806000001000900 # 4-2-4-1-1
004000900000604000203000705070806010000020000020905080605000807000507000008000300 # 4-2-4-2-1
001000900000264000200000005060090070080602090050070010800000004000425000009000100 # 4-3-0-1-0
001000300000967000400000006070040090050293010040050020500000008000638000003000200 # 4-3-0-1-1
003000200000964000100000008050308040080000020010409030600000009000153000008000400 # 4-3-0-2-0
003000800000418000400000007030209070020040030010503080100000004000361000009000600 # 4-3-0-2-1
005000600000894000400010009060000050034050970010000030600070002000639000001000500 # 4-3-1-0-1
008000400003060700240000013000602000080000090000704000790000056001020900004000300 # 4-5-0-2-0
002000900010000040400802007004050600000301000003080200200709001090000030005000400 # 4-8-2-1-0
002000900030000040500207006006701500000000000001502400900306001080000060004000200 # 4-8-2-2-0
002000700080000090600809001007508200000030000008201400900407008050000070004000900 # 4-8-2-2-1
008000300030050080709000506000706000010000040000902000903000407070080090002000800 # 4-9-4-2-0
001000400040507080600010005010000070008000300020000090400060009030401050009000800 # 4-10-1-0-0
001000200045000780370000061000183000000409000000527000680000042013000950007000600 # 4-12-0-3-0
005000600014060890380000042000702000030000050000401000150000024043010780007000300 # 4-13-0-2-0
003060900000708000600000004070080010200406009090010020700000001000305000004090300 # 5-2-0-1-0
001030600000108000700000001090802070400000009020601080800000005000903000006070100 # 5-2-0-2-0
008070200000203000600090003040000070709000508030000090100030007000905000004060800 # 5-2-1-0-0
001060300000203000800050002070000080504030601030000070700020004000105000006090800 # 5-2-1-0-1
006020700000406000500803006053000140400000007029000650200304009000708000008090500 # 5-2-2-0-0
001070900030000050804000102000106000300000008000409000108000407040000020006020800 # 5-8-4-2-0
007030200040906030500000004060000050100000006080000040900000007030502080008060500 # 5-10-0-0-0
002308700000000000700020005600709004004000300900601008800060003000000000001402500 # 6-0-1-2-0
006708300000000000800412006102000903007000800608000705500146009000000000004905600 # 6-0-3-0-0
001908300000000000803000204200030007000802000100070005509000108000000000004501900 # 6-0-4-1-0
007208100000070000300000005800504006060000090100307004700000008000050000005801200 # 6-1-0-2-0
003701600000090000500020007400000002032000410900000006300070005000050000006208300 # 6-1-1-0-0
003201500000040000500306007206000901030000020801000405100402003000060000005803100 # 6-1-2-0-0
001705600000301000800000002730000094000000000260000035400000007000609000005207400 # 6-2-0-0-0
008109300000607000100000008620000017000060000840000026400000005000301000005204100 # 6-2-0-0-1
001504900020000060500000004900080006000201000300070001200000005060000070003807400 # 6-8-0-1-0
008209700060000080300000005500080001000624000800030006600000007050000040002506800 # 6-8-0-1-1
001205300040000070200000006700302008000000000400906005600000002030000040007603900 # 6-8-0-2-0
007802400040000060200000001800407002000090000900508003400000007080000090002603800 # 6-8-0-2-1
010000050400000002000719000007020100004607300006030400000276000700000009030000040 # 8-0-3-1-0
030000050900000006000615000001090800009537600002040300000763000400000008010000090 # 8-0-3-1-1
010000070900000008000587000008401300003000700001703200000894000200000005060000010 # 8-0-3-2-0
070000060300000005001907400008040600000602000009050300007308900800000001030000080 # 8-0-6-1-0
030000070900000006006307400002908100000000000007504300001406200300000005020000080 # 8-0-6-2-0
010000050300050009000807000007010900050203060009070800000604000400030008090000020 # 8-1-2-1-0
030000010900030006000809000007060900020147030004050600000302000800010003060000070 # 8-1-2-1-1
030000050500080007000302000007601400090000060006208700000705000800030001020000090 # 8-1-2-2-0
050000080900702004000090000090040070001609500080070020000010000200306001070000050 # 8-2-1-1-0
050000040600204001000801000026000750000000000075000830000105000400302008010000090 # 8-2-2-0-0
020000030300407008008000400060080020000302000050060010009000100200503006010000070 # 8-2-4-1-0
010000040200407005009000700040010030000862000070050060003000900500103008080000020 # 8-2-4-1-1
070000080205000301060040050000501000007060100000402000010020040608000502050000060 # 8-4-1-2-1
040000020701000308065000710000692000000507000000184000072000480304000107080000060 # 8-4-4-3-0
010000080370000091000407000008040900000908000002050100000603000950000024030000010 # 8-8-2-1-0
060000070820000061000801000004030200000725000005090600000903000570000028090000030 # 8-8-2-1-1
040000070920000061000603000004502900000000000008309700000805000270000043060000010 # 8-8-2-2-0
030000040940000061000509000002104500000080000006705200000401000560000097010000020 # 8-8-2-2-1
010030080500000003000208000005040700300601009006090400000809000900000001070010040 # 9-0-2-1-0
040070050800000002000601000007305600600000009009204300000506000300000004020080070 # 9-0-2-2-0
050070030900000006000304000009103400800060002001209800000701000600000004040090020 # 9-0-2-2-1
010020070400010006000305000001000300320000089004000500000204000600030002090070060 # 9-1-2-0-0
060090050700506001000000000010709060800030005070805040000000000100603004020040030 # 9-2-0-2-1
080040020400107008000050000090000080203060504070000030000070000600203001020010090 # 9-2-1-0-1
030050090460000031007000200000703000900040006000206000005000100670000058040080020 # 9-8-4-2-1
060804050100000007000020000200508009004000700900201004000080000800000002070605010 # 10-0-1-2-0
010603080400000006000208000607000308000000000308000705000309000700000002030405090 # 10-0-2-0-0
070309020300000005000801000503000602000090000106000407000706000800000001010902060 # 10-0-2-0-1
010708050900000007004000600800070001000304000300080004009000700500000008020105060 # 10-0-4-1-0
050209040200000001008000700100020006000685000800030009009000200300000004040708060 # 10-0-4-1-1
020806090500000004009000600300601002000030000800705009005000300200000008040509020 # 10-0-4-2-1
090601020600080005000000000400030009060805070200070003000000000800050004010903050 # 10-1-0-1-0
080106030900502001000000000430000062000020000210000078000000000800701004060405080 # 10-2-0-0-1
070805090380000072000000000200080005000704000800010007000000000640000029050108030 # 10-8-0-1-0
050908070910703048000000000630000084000000000180000063000000000560401037040307010 # 10-10-0-0-0
028000750400060003100000004000502000090000040000401000900000005200030006045000830 # 12-1-0-2-0
400000008000508000000693000048000130009000500051000980000975000000201000700000006 # 16-2-3-0-0
400000006000408000006010200090070080004603900030020010002040100000507000300000008 # 16-2-5-1-0
200000009000402000004301500039000650000000000076000320007906100000503000100000008 # 16-2-6-0-0
300000005000708000008403200015000670000080000079000830001805300000104000200000007 # 16-2-6-0-1
500000004000247000006000900030090010060408030050070040009000500000831000800000001 # 16-3-4-1-0
900000008000468000001000600080609010040000030020305090002000900000281000500000004 # 16-3-4-2-0
100000002008000500090208070005030600000104000004020300030905080006000200400000005 # 16-4-2-1-0
300000008008000900010502060007405100000000000005906400030604010006000500800000003 # 16-4-2-2-0
200000004007000100080903050008201600000030000005609300020805010009000700100000005 # 16-4-2-2-1
600000008008070400070106030005000300040000050009000600050201070007040800900000003 # 16-5-2-0-0
200000003009504200010000040060080020000205000080090070070000010001603500300000004 # 16-6-0-1-0
400000007007206100050000080080020060000967000020080090090000050008309200200000001 # 16-6-0-1-1
400000007006907200020000040040201090000000000080703060090000020001605400300000008 # 16-6-0-2-0
600000001070000080000628000007040600009302100008090300000436000020000050800000009 # 16-8-3-1-0
500000009010000080000457000001060700006732100009010300000985000080000020700000003 # 16-8-3-1-1
300000006020000090001405700007030200000806000003010500004107800070000040500000009 # 16-8-6-1-0
100000005060000090005604100007060900000253000002070400006501800040000010200000003 # 16-8-6-1-1
200000008090030020000104000004060700020307060005040100000503000050090040700000001 # 16-9-2-1-0
100000006080060020000809000005902100010000080003506200000705000070020030900000004 # 16-9-2-2-0
400000001020706040000203000013000650000000000045000980000807000030405060600000002 # 16-10-2-0-0
200000003080501020003000700070010030000209000010040060001000800060107040500000009 # 16-10-4-1-0
800000007030508010007010900080000090003000600050000070002080500040306080100000004 # 16-10-5-0-0
100000004072000390090050070000207000003000800000601000080090010026000580700000009 # 16-12-1-2-0
100090005000305000009000800090030040600802003080070050005000700000706000200010009 # 17-2-4-1-0
900060001000307000007090500010000030604000802090000060002050900000401000100080003 # 17-2-5-0-0
400050003003000400080020050000506000906000108000701000090070080001000300200030005 # 17-4-1-2-0
500030008020000050000509000009040600100603009003010700000401000060000090800070001 # 17-8-2-1-0
600030005090000030000109000001906300700000004003804100000403000020000070500020006 # 17-8-2-2-0
600010009097000530080000070000208000500000004000309000040000010032000740900020008 # 17-12-0-2-0
500609002000010000006502300907000104040000020105000706009203500000090000600408003 # 18-1-6-0-0
100502009000604000005000400960000027000000000570000041008000300000705000600103005 # 18-2-4-0-0
700201008005000600040000090400080001000305000500020003090000060004000100300902005 # 18-4-0-1-0
600109007030000020000080000100706008009000600700803004000050000040000060800907003 # 18-8-1-2-0
900108003020000070007000900500060004000501000700020001003000400010000060200307009 # 18-8-4-1-0
402000608000000000900506003001060400000107000008050300200809004000000000604000705 # 20-0-2-1-0
409000102000607000600000009080010090000806000040050010200000008000304000807000601 # 20-2-0-1-0
509000702000801000100000003020904050000000000040305060300000008000106000804000601 # 20-2-0-2-0
703000209060000070400030001000405000001000700000809000300040002050000090207000508 # 20-8-1-2-0
309000701020040050400000003000509000050000010000701000800000006070080030204000809 # 20-9-0-2-0
206000304040060050800000002000508000020030070000704000600000009010090040903000508 # 20-9-0-2-1
710000068800000009000703000002090800000607000006010500000901000500000001980000024 # 24-0-2-1-0
150000074300000009000408000005040300000516000007090400000209000700000001280000096 # 24-0-2-1-1
390000065700000008000702000005307800000000000003408900000504000100000004470000021 # 24-0-2-2-0
350000049100000002004050300000603000007000600000102000005010400900000007270000095 # 24-0-5-2-0
760000098100050007005000100000102000080000030000903000001000300800060004970000056 # 24-1-4-2-0
340000057500102009000000000020080070000206000010040020000000000800409001650000038 # 24-2-0-1-0
780000034500804006000000000090050070000412000010060050000000000200706008630000015 # 24-2-0-1-1
270000091608000507090000060000315000000209000000648000030000070406000908510000034 # 24-4-0-3-0
gsf
2014 Supporter

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

JPF wrote:24- 0- 1- 2- 0 ; 20 clues ; SM (symmetric minimal)
Is it possible to make it absolutely minimal ?
JPF

Hi JPF

I like the idea of creating a database of fully symmetric puzzles..

But can you explain your terminology of minimality to me...

SM = symmetric minimal means ... I am guessing that you mean it is minimal in the sense that removing any ENTIRE ORBIT (thus preserving the symmetry) makes it no longer valid

AM = absolutely minimal means ... removing any INDIVIDUAL CELL makes it no longer valid.. i.e. minimal in the usual sense.

So the perfect puzzle would be fully symmetric and absolutely minimal..

Is this the right idea...

Gordon
gfroyle

Posts: 214
Joined: 21 June 2005

i think he has the right idea anyone else
nathanmcmaster

Posts: 5
Joined: 19 April 2006

Here's a minimal version of the 20 clue square puzzle:
Code: Select all
`38.|...|.719..|...|..4...|.2.|...-----------...|8.9|.....4|...|6.....|1.2|...-----------...|.5.|...8..|...|..773.|...|.98`

As well as 24-0-1-2-0, it can be thought of as 20-1-0-2-0, or 8-8-1-2-0 or 4-1-4-0-0
or 1-12-0-2-0 or 1-4-4-2-0. Generally every fully symmetric puzzle has 6 ways it can be written
corresponding to swapping rows 1,2 and 3 and columns 1,2 and 3 (as well as rows 7,8 and 9 and columns
7,8 and 9). I suggest you use the lowest number to record it, unless you want to count all 6 puzzles
even though they're esentially the same sudoku puzzle.

some more minimal 20s:
Code: Select all
`.3.|...|.4.6..|...|..3..7|.2.|9..-----------...|3.8|.....9|...|7.....|5.6|...-----------..2|.1.|4..5..|...|..6.9.|...|.7.`

from the superior thread. 8-0-5-2-0 but also 4-9-0-2-0 and 17-4-0-2-0. This has only 3 representations because
swapping rows 1 and 2 and columns 1 and 2 (along with rows 8 and 9 and columns 8 and 9) doesn't alter the pattern.
Code: Select all
`6..|...|..5...|4.3|.....5|.7.|1..-----------.2.|...|.7...7|...|2...8.|...|.3.-----------..6|.2.|5.....|7.8|...4..|...|..3`

16-2-5-0-0
2-8-5-0-0
17-2-4-0-0
16-9-2-0-0
2-9-4-0-0
17-8-2-0-0
Code: Select all
`...|7.2|.....2|.4.|6...6.|...|.5.-----------4..|...|..9.1.|...|.3.3..|...|..7-----------.8.|...|.9...4|.2.|1.....|5.3|...`

2-5-0-0-0
5-2-0-0-0
9-0-2-0-0
8-1-2-0-0
4-2-1-0-0
2-4-1-0-0

Code: Select all
`...|...|.....2|5.8|1...7.|...|.9.-----------.5.|.6.|.4....|3.2|....6.|.9.|.3.-----------.9.|...|.7...8|2.1|3.....|...|...`

0-4-2-1-0
0-6-0-1-0
4-0-2-1-0
6-0-0-1-0
8-2-0-1-0
10-0-0-1-0

Code: Select all
`...|2.5|.....7|...|4...9.|...|.3.-----------2..|.7.|..1...|8.3|...1..|.9.|..5-----------.4.|...|.8...8|...|5.....|7.1|...`

2-4-0-1-0
4-2-0-1-0
8-0-2-1-0
ab

Posts: 451
Joined: 06 September 2005

gsf wrote:here is a list of minimal full symmetric puzzles generated overnight
sorted and uniq'd by your nomenclature...

Thanks gsf.
That’s a lot of work for a night…

I’m trying to organize myself to list the puzzles.
I noticed 3 “popular” patterns, already posted by Ocean in this thread :
2-4-1-1-0
6-8-0-2-0
8-8-2-1-1
absolutely minimal (M) in each case…

and 3 circles (radius = 4)
3-4-0-1-0
3-4-0-2-0
3-5-0-0-1

One of them :
3- 4- 0- 2- 0 ; 24 clues ; M
Code: Select all
` . . . | 2 3 5 | . . . . . 9 | . . . | 8 . . . 3 . | . . . | . 1 .-------+-------+------- 8 . . | 4 . 7 | . . 2 1 . . | . . . | . . 4 4 . . | 5 . 2 | . . 6-------+-------+------- . 7 . | . . . | . 5 . . . 1 | . . . | 6 . . . . . | 9 8 6 | . . .`

No 20-21 clues for the moment …

gfroyle wrote:But can you explain your terminology of minimality to me...

SM = symmetric minimal means ... I am guessing that you mean it is minimal in the sense that removing any ENTIRE ORBIT (thus preserving the symmetry) makes it no longer valid
AM = absolutely minimal means ... removing any INDIVIDUAL CELL makes it no longer valid.. i.e. minimal in the usual sense.

So the perfect puzzle would be fully symmetric and absolutely minimal..
Is this the right idea...
Yes, you are right.

I will try to develop :
A proper puzzle P is minimal if, for every puzzle Q such that Q<P (the set of clues of Q is included in the set of clues of P) ,
then N(Q)>1 ; (number of solutions of Q).

We can extend the definition of minimality to a class of puzzles.

Let s be a symmetry of the square (like you describe them).
A puzzle P is s-symmetric if its pattern P*={A1,…, An}is such that :
if Ai € P* => s(Ai) € P* ; i=1,..., n(P), number of clues of P.
A s-puzzle P will be s-minimal if, for every s-puzzle Q such that Q<P, then N(Q)>1.

And extend again this concept for a set of symmetries : s1, s2,…, sp :

A puzzle P is s1,s2,…,sp-symmetric if its pattern P*={A1,…, An}is such that :
if Ai € P* => sk(Ai) € P* ; i=1,...,n(P) ; k=1,...,p.
A s1,s2,…,sp - puzzle P will be s1,s2,…,sp - minimal if, for every s1,s2,…,sp -puzzle Q such that Q<P, then N(Q)>1.

Here we are considering the puzzles having the dihedral group as symmetries.
The puzzles Q are created by removing at least one orbit from P (i.e. : 1, 4 or 8 clues) to preserve the symmetry.
It’s what we will call symmetric-minimal puzzle(SM).

Obviously, an (absolutely) minimal (M) puzzle is a symmetric-minimal (SM) puzzle.

We can note that all the 20 clues fully symmetric puzzles are SM :
1. Each orbit has at least 4 clues.
2. There is no 16 clues proper puzzle (?)

gfroyle wrote:So the perfect puzzle would be fully symmetric and absolutely minimal..

I agree.

JPF
JPF
2017 Supporter

Posts: 6001
Joined: 06 December 2005
Location: Paris, France

JPF wrote:We can note that all the 20 clues fully symmetric puzzles are SM

Yes. By accepting the conjecture that no symmetrical 17 exists, we can conclude that 20 is the minimum number for a fully symmetric puzzle, and also that a 21 clues fully symmetric is SM as long as removing the centre clue no longer makes it valid.
kjellfp

Posts: 140
Joined: 04 October 2005

Thanks ab for these 30 fully symmetric and minimal 20 clues puzzles.
ab wrote:As well as 24-0-1-2-0, it can be thought of as 20-1-0-2-0, or 8-8-1-2-0 or 4-1-4-0-0
or 1-12-0-2-0 or 1-4-4-2-0. Generally every fully symmetric puzzle has 6 ways it can be written
corresponding to swapping rows 1,2 and 3 and columns 1,2 and 3 (as well as rows 7,8 and 9 and columns
7,8 and 9).

Earlier in this thread, Ruud made the same comment, that I still have in mind :
Ruud wrote:Have you considered that simultaneous permutation of rows 1-3 & 7-9 and columns 1-3 & 7-9 allow you to reduce the number of patterns that you need to find?

To illustrate what you (and Ruud) mentioned, let’s have a look on the different patterns generated with your 20 clue square puzzle :
Code: Select all
`24-0-1-2-0 3 8 . | . . . | . 7 1 9 . . | . . . | . . 4 . . . | . 2 . | . . .-------+-------+------- . . . | 8 . 9 | . . . . . 4 | . . . | 6 . . . . . | 1 . 2 | . . .-------+-------+------- . . . | . 5 . | . . . 8 . . | . . . | . . 7 7 3 . | . . . | . 9 88-8-1-2-0 . 9 . | . . . | . 4 . 8 3 . | . . . | . 1 7 . . . | . 2 . | . . .-------+-------+------- . . . | 8 . 9 | . . . . . 4 | . . . | 6 . . . . . | 1 . 2 | . . .-------+-------+------- . . . | . 5 . | . . . 3 7 . | . . . | . 8 9 . 8 . | . . . | . 7 .1-4-4-2-0 . . . | . 2 . | . . . . . 9 | . . . | 4 . . . 8 3 | . . . | 1 7 .-------+-------+------- . . . | 8 . 9 | . . . 4 . . | . . . | . . 6 . . . | 1 . 2 | . . .-------+-------+------- . 3 7 | . . . | 8 9 . . . 8 | . . . | 7 . . . . . | . 5 . | . . .20-1-0-2-0   3 . 8 | . . . | 7 . 1 . . . | . 2 . | . . . 9 . . | . . . | . . 4-------+-------+------- . . . | 8 . 9 | . . . . 4 . | . . . | . 6 . . . . | 1 . 2 | . . .-------+-------+------- 8 . . | . . . | . . 7 . . . | . 5 . | . . . 7 . 3 | . . . | 9 . 84-1-4-2-0 . . 9 | . . . | 4 . . . . . | . 2 . | . . . 8 . 3 | . . . | 1 . 7-------+-------+------- . . . | 8 . 9 | . . . . 4 . | . . . | . 6 . . . . | 1 . 2 | . . .-------+-------+------- 3 . 7 | . . . | 8 . 9 . . . | . 5 . | . . . . . 8 | . . . | 7 . .1-12-0-2-0 . . . | . 2 . | . . . . 3 8 | . . . | 7 1 . . 9 . | . . . | . 4 .-------+-------+------- . . . | 8 . 9 | . . . 4 . . | . . . | . . 6 . . . | 1 . 2 | . . .-------+-------+------- . 8 . | . . . | . 7 . . 7 3 | . . . | 9 8 . . . . | . 5 . | . . .`

That is everything from the square... to the circle !

Let’s say that these puzzles and therefore these patterns, are S-equivalent.

By opening this thread, my initial idea was to list all the FS (fully symmetric) patterns and for each of them to give (if possible) a valid minimal puzzle.
It is true that a FS puzzle (with its clues) gives, in addition, up to 5 new FS puzzles with different patterns.
Obviously, these adequate permutations of rows and columns preserve all type of minimality (and the number of clues !)
ab wrote:I suggest you use the lowest number to record it, unless you want to count all 6 puzzles
even though they're esentially the same sudoku puzzle.

As I don’t know yet how I’m going to list all this examples of valid patterns, I would suggest that we post one of the puzzles, the code of its pattern and if possible, the codes of the S-equivalent patterns.
Exactly as you did ab.

JPF
JPF
2017 Supporter

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Joined: 06 December 2005
Location: Paris, France

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