The New Sudoku Players' Forum

[dobrichev]

I will check for which of the 475 puzzles the Glenn's tool didn't find symmetrical representation. That should be the reason they are missed in the "niced" form.
Actually they must have kinds of these "lesser known" symmetries that we are now interested in. Right?

[blue]

Actually they must have kinds of these "lesser known" symmetries that we are now interested in. Right?

There are some with 3 automorphisms. This one has "jumping diagonals" symmetry.

Code: Select all

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

Earlier in the thread, there were two with multiple symmetries -- double diagonal and pi/sticks.
I think this one is the only other one with multiple symmetries -- jumping diagonals + pi rotational (6 automorphisms).

Code: Select all

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

There are 6 with quarter turn symmetry.

Code: Select all

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

[eleven]

I manually transformed the 12 sticks symmetric puzzles in the 36symmetric file to normal form - lines 4,5,12,13,14,15,16,23,24,41,42,201 (one still missing in nice form ?).
But i saw now, that in the 37 symmetric niced list also 20 puzzles are missing.

Hidden Text: Show

Code: Select all

 +-------+-------+-------+
 | . 2 4 | 9 . 5 | . . 1 |
 | 6 5 . | . . 2 | 9 . . |
 | 9 . 1 | 4 6 . | . . 2 |
 +-------+-------+-------+
 | . . . | 5 4 6 | 8 . 9 |
 | . . . | 8 . 9 | . 1 . |
 | . . . | . . . | . . . |
 +-------+-------+-------+
 | 4 3 . | 6 . 8 | 1 . . |
 | . 6 5 | 3 . . | . . 8 |
 | 1 . 8 | . 5 4 | 3 . . |
 +-------+-------+-------+ #1 col.sticks
 +-------+-------+-------+
 | 3 . 8 | . 5 4 | . . 7 |
 | 4 6 . | 3 . 7 | . . 8 |
 | . . 5 | 6 8 . | 3 . . |
 +-------+-------+-------+
 | 2 . 3 | . . . | . 7 . |
 | 5 4 6 | . . . | 2 . 3 |
 | . . . | . . . | . . . |
 +-------+-------+-------+
 | 9 . 2 | 4 6 . | 7 . . |
 | . 5 4 | 7 . 2 | 9 . . |
 | 6 . . | . 9 5 | . . 2 |
 +-------+-------+-------+ #2 col.sticks
 +-------+-------+-------+
 | 2 . . | 4 . . | 7 8 . |
 | . 4 7 | 1 . . | . . 6 |
 | 8 6 . | . . 2 | . 1 4 |
 +-------+-------+-------+
 | 6 . 8 | 3 . 4 | 1 7 5 |
 | . . . | . 2 . | 6 . 8 |
 | . . . | . . . | . . . |
 +-------+-------+-------+
 | . . 2 | . . 3 | . 6 7 |
 | 7 3 . | . . 5 | 8 . . |
 | . 8 6 | 2 . . | 3 5 . |
 +-------+-------+-------+ #3 col.sticks
 +-------+-------+-------+
 | . . 2 | . . 9 | 5 4 . |
 | . . . | 2 . . | . . . |
 | 9 8 . | . . 5 | . . 2 |
 +-------+-------+-------+
 | . 9 1 | 7 . 4 | 6 . . |
 | 7 . 8 | . . 6 | 1 . 4 |
 | 6 . . | 1 . 8 | . 5 7 |
 +-------+-------+-------+
 | 2 . 7 | . . 1 | . 6 . |
 | . 1 9 | . . . | . 7 5 |
 | . 6 . | . . 7 | 2 . 1 |
 +-------+-------+-------+ #4 row sticks
 +-------+-------+-------+
 | 1 2 . | . 8 . | 4 . 6 |
 | . 9 . | . . . | . 3 . |
 | 4 . 6 | . 2 . | 7 8 . |
 +-------+-------+-------+
 | . 8 9 | . 1 2 | 6 . 7 |
 | 2 . . | . 6 . | 8 . . |
 | 6 . 1 | . 7 8 | . 2 3 |
 +-------+-------+-------+
 | . 1 . | . 9 . | 3 . 4 |
 | . . . | . . 4 | . . . |
 | 9 . 4 | . 3 . | . 7 . |
 +-------+-------+-------+ #5 row sticks
 +-------+-------+-------+
 | 4 . . | . 9 8 | . 3 1 |
 | 3 . . | . . 4 | 9 . . |
 | . 9 7 | . 3 2 | 4 . . |
 +-------+-------+-------+
 | 2 . 3 | . . 9 | 5 4 . |
 | . 7 9 | . . . | . 1 3 |
 | 5 4 . | . . 3 | 8 . 9 |
 +-------+-------+-------+
 | 7 5 . | . . 1 | . . 8 |
 | . . . | . 5 . | . . . |
 | . . 2 | . . 7 | 1 5 . |
 +-------+-------+-------+ #6 row sticks
 +-------+-------+-------+
 | 1 2 . | . 8 . | 4 . 6 |
 | . 9 8 | . . . | . 3 2 |
 | 4 . 6 | . 2 . | 7 8 . |
 +-------+-------+-------+
 | . 8 9 | . 1 2 | 6 . . |
 | 2 . . | . 6 . | 8 . . |
 | 6 . . | . 7 8 | . 2 3 |
 +-------+-------+-------+
 | . 1 . | . 9 . | 3 . 4 |
 | . . . | . . 4 | . . . |
 | 9 . 4 | . 3 . | . 7 . |
 +-------+-------+-------+ #7 row sticks
 +-------+-------+-------+
 | 1 2 . | . . . | . . 9 |
 | . 9 7 | . . . | . 6 4 |
 | . . 6 | . . . | 1 2 . |
 +-------+-------+-------+
 | 2 . . | 9 . 8 | . 4 6 |
 | 6 . 8 | . . 2 | 9 . 5 |
 | . 7 9 | 6 . 5 | 2 . . |
 +-------+-------+-------+
 | 7 8 . | 5 . 4 | . . 1 |
 | . . . | 1 . . | . . . |
 | . . 1 | 8 . 7 | 4 5 . |
 +-------+-------+-------+ #8 row sticks
 +-------+-------+-------+
 | 1 . . | . . . | . 5 6 |
 | 7 . 8 | . . . | 1 . 2 |
 | . 5 6 | . . . | 7 . . |
 +-------+-------+-------+
 | 5 . . | 8 9 . | . 2 3 |
 | . . . | . 5 . | . . . |
 | . 8 9 | 2 3 . | 5 . . |
 +-------+-------+-------+
 | . 6 . | 9 7 . | 2 . 1 |
 | 9 1 . | 6 . . | 3 7 . |
 | 8 . 7 | 3 1 . | . 6 . |
 +-------+-------+-------+ #9 row sticks
 +-------+-------+-------+
 | . 5 6 | 7 . . | 1 . 3 |
 | 1 2 . | . . 6 | 7 8 . |
 | 7 . 9 | 1 . . | . 5 6 |
 +-------+-------+-------+
 | 2 1 . | . 7 . | . 3 5 |
 | . . 7 | 5 . . | . . 1 |
 | . 9 5 | . 1 . | 8 7 . |
 +-------+-------+-------+
 | . 6 2 | . 9 . | 3 . . |
 | . . . | . . . | . . . |
 | 9 . . | . 3 . | . 6 8 |
 +-------+-------+-------+ #10 row sticks
 +-------+-------+-------+
 | 4 . 6 | . . 3 | 7 8 . |
 | . 3 2 | 7 . . | . 6 5 |
 | 7 8 . | . . 6 | 1 . 3 |
 +-------+-------+-------+
 | 8 4 . | . 3 . | . 5 6 |
 | 6 . . | . . 8 | 3 . . |
 | . 2 3 | . 6 . | 8 1 . |
 +-------+-------+-------+
 | 2 7 . | . 4 . | . . 1 |
 | . . . | . . . | . . . |
 | . . 4 | . 1 . | 5 7 . |
 +-------+-------+-------+ #11 row sticks
 +-------+-------+-------+
 | . . 7 | . 8 . | 9 . . |
 | 6 . 8 | 9 . . | 7 3 1 |
 | . . . | 1 6 . | . 8 2 |
 +-------+-------+-------+
 | 2 . 3 | 5 . 6 | . . . |
 | 5 . 6 | . . . | 2 1 3 |
 | . 7 . | . . . | . . . |
 +-------+-------+-------+
 | 7 . . | . 9 . | . . 8 |
 | 9 . 5 | . . 8 | 1 2 7 |
 | . . . | . 5 1 | 3 9 . |
 +-------+-------+-------+ #12 col. sticks

[dobrichev]

There are exactly 12 missing 36s in the nice form (463 + 12 = 475). The last one is not on line 201 but on 202. Thanks.

Blue, I feel that your CPU is hot now.

Soon or later some formal definition of "best look" for any kind of symmetry should appear. I am expecting half ton of examples from the most canonical grid.

[champagne]

There are some with 3 automorphisms. This one has "jumping diagonals" symmetry.

Earlier in the thread, there were two with multiple symmetries -- double diagonal and pi/sticks.
I think this one is the only other one with multiple symmetries -- jumping diagonals + pi rotational (6 automorphisms).

I have to confess that I have to go to the original symmetry link to understand the "jumping diagonals" symmetry.

regarding the second sentence, I just add that the Rotational 90 is a native "multiple symmetry".

[eleven]

I have to confess that I have to go to the original symmetry link to understand the "jumping diagonals" symmetry.

champagne,

move the stacks cyclically right and the bands down. Then box 1 moves to box 5, 5 to 9 and 9 to 1, 4->8->3->4, 7->2->6->7.
So in blues' nice puzzle you have
1896 in box 1 go to 2974 in box5, these to 3785 in box9,these back to box 1.
81 in box 4 -> 92b8->73b3,
698231b7->479312b2->587123b6.

[champagne]

eleven,
thanks for that short and clear description of the symmetry

[Mauricio]

I am glad there is new interest in symmetric puzzles, now in the high-clue puzzles. To understand more about those "rare" symmetries, perhaps it would be useful to read down-under-upside-down-a-sudoku-puzzle-t6355-75.html#p64593 and the posts after it, in fact, the whole thread is very interesting.

[Leren]

FWIW I've produced a single text file that contains dobrichev's 36 - 38 clue niced puzzles in line format.

Leren

PS.If you haven't downloaded it yet, the file also includes, for each puzzle, the sequence number and dobrichev's description of the expected symmetry.

Leren

[eleven]

There are exactly 12 missing 36s in the nice form (463 + 12 = 475).

The problem with the count came from the number "4" (automorphisms) for the niced puzzle nr 392. In fact all have 2 automorphisms and, as mentioned earlier, gsf's "da" (diagonal/antidiagonal) for #436 is a bug (with double diagonal symmetry there must be at least 4 automorphisms).

[blue]

regarding the second sentence, I just add that the Rotational 90 is a native "multiple symmetry".

I don't think so. The automorphism group has the pi-rotational group as a proper subgroup, for sure, but can you name another of the "multiple symmetries" ? At minimum, an acceptable answer should correspond to another non-trivial proper subgroup, and there are none to choose from.

[blue]

Blue, I feel that your CPU is hot now.

It was, over the weekend
More to report on in a few days ...

[eleven]

regarding the second sentence, I just add that the Rotational 90 is a native "multiple symmetry".

I don't think so. The automorphism group has the pi-rotational group as a proper subgroup, for sure, but can you name another of the "multiple symmetries" ? At minimum, an acceptable answer should correspond to another non-trivial proper subgroup, and there are none to choose from.

It is not easy to say, what is what here or not. It depends on background, definitions, terminology and intentions. See the old discussion here.

[Added:] E.g. note the slight difference between a puzzle with jumping diagonals + pi rotational symmetry (as blue has provided here) and one with jumping diagonals + diagonal symmetry. Both have 2 symmetries and 6 automorphisms, but only the second has an automorphism with number cycles of length 6 (and therefore an own "conjugacy class").

Code: Select all

    +-------+-------+-------+
    | 3 . . | 5 . 8 | 7 2 1 | 
    | . . . | . . . | . . . |
    | . . 8 | 7 9 1 | 3 . 6 |
    +-------+-------+-------+
    | 8 3 2 | 1 . . | 6 . 9 |
    | . . . | . . . | . . . |
    | 1 . 4 | . . 9 | 8 7 2 |
    +-------+-------+-------+
    | 4 . 7 | 9 1 3 | 2 . . |
    | . . . | . . . | . . . |
    | 9 8 3 | 2 . 5 | . . 7 |
    +-------+-------+-------+ H (1,9)(2,8)(3,7),(4,6)(5) + JD (123),(456),(789)
   
    +-------+-------+-------+
    | 3 . . | 2 . . | 1 . 4 |
    | . . . | . 1 7 | . 3 . |
    | . . . | 5 . . | . 6 . |
    +-------+-------+-------+
    | 2 . 8 | 1 . . | 3 . . |
    | . 1 . | . . . | . 2 5 |
    | . 4 . | . . . | 9 . . |
    +-------+-------+-------+
    | 1 . . | 3 . 6 | 2 . . |
    | . 3 9 | . 2 . | . . . |
    | 7 . . | . 8 . | . . . |
    +-------+-------+-------+ D (1)(2)(3)(47)(58)(69) + JD (123)(486)(579)
    DBS (diagonal mirror, bands down, stacks right): (123)(456789)

[Serg]

Hi, all!
Just thinking about 2 ways of symmetry concepts - the first one is "human", based on geometric concepts, and second one is based on Group Theory concepts. It would be nice to find common approach ...

Let's consider solution grids only. It simplifies symmetry analysis.

First, I should say, that half turn (pi rotation) symmetry always come with double diagonal symmetry (and with vertical+horizontal symmetry, and with quarter turn symmetry).

Double diagonal symmetry (DDS) is represented by automorphisms group of order 4 (it contains 4 VPTs). Quarter turn symmetry (QTS) is represented by automorphisms group of order 4 too. What symmetry is "more symmetric"?

DDS automorphisms group contains 3 non-trivial subgroups (having 2 VPTs each), representing 3 "basic" simmetries - diagonal, antidiagonal and half turn symmetries. All its VPTs participate one of 3 subgroups. So, the group itself is not primitive, but "composed" of other basic symmetries. Totally, DDS automorphisms group contains 3 "basic" symmetries.

QTS automorphisms group contains only 1 non-trivial subgroup - half turn symmetry. 2 (out of 4) VPTs don't participate this subgroup, so we should count automorphisms group itself as "basic" symmetry. Totally, QTS automorphisms group contains 2 "basic" symmetries.

Therefore double diagonal symmetry is more symmetric than quarter turn symmetry.

Serg

[blue]

regarding the second sentence, I just add that the Rotational 90 is a native "multiple symmetry".

I don't think so. The automorphism group has the pi-rotational group as a proper subgroup, for sure, but can you name another of the "multiple symmetries" ? At minimum, an acceptable answer should correspond to another non-trivial proper subgroup, and there are none to choose from.

I've probably been too quick to say "no" ... maybe subconsciously wanting to defend my "second sentence".

Totally, QTS automorphisms group contains 2 "basic" symmetries

I can't argue that, for example.

(...)

Therefore double diagonal symmetry is more symmetric than quarter turn symmetry.

Maybe it's true
Here's a puzzle that can have either, but not both at the same time.
Does one "look more symmetric" than the other ?

Code: Select all

2 3 . . . 7 . . 4                 . . 4 . . 7 2 3 .
. . 7 . . 4 . . 1                 . . 1 . . 4 . . 7
. . 4 . . 1 8 9 .                 8 9 . . . 1 . . 4
1 2 3 . . . . . .   swap stacks   . . . . . . 1 2 3
. . . . . . . . .     1 and 3     . . . . . . . . .
. . . . . . 7 8 9   <--     -->   7 8 9 . . . . . .
. 1 2 9 . . 6 . .                 6 . . 9 . . . 1 2
9 . . 6 . . 3 . .                 3 . . 6 . . 9 . .
6 . . 3 . . . 7 8                 . 7 8 3 . . 6 . .

Note: column and row stick symmetries, are also present.
I think it's the only (truly) minimal puzzle of its kind.
Too bad it's solvable with singles.