The New Sudoku Players' Forum

[marek stefanik]

Thank you for your solutions.
As pointed out first by Bogdan, the permutation rows|columns = 987654321|987456321, digits = 123547698 results in the same puzzle, only with r19c456 altered.
Since those contain the same digits, we can swap them back and make the same inferences as with a full-grid automorphism.

I found it by playing with the sticks symmetry, I was curious if I could swap two of the bands instead of just swapping the rows within each of them.
This symmetry breaks in the swapped sticks, as they must contain the same digits in the same columns, but the (fully-given) DP solves this issue.

The idea of combining fully-given DPs with GSP is not new per se (see this puzzle created by eleven), but I hadn't seen it used to allow for a symmetry that doesn't have a full solution grid.

I have created this bonus puzzle (9.2 skfr), showcasing a very similar symmetry:
..........52.3.1.74.1.2.36.7..9...8..9...2.34..43.....8..2...53.6...8..9..5..3...

Marek

[eleven]

[see below]

[eleven]

I have created this bonus puzzle (9.2 skfr), showcasing a very similar symmetry:
..........52.3.1.74.1.2.36.7..9...8..9...2.34..43.....8..2...53.6...8..9..5..3...

Code: Select all

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

Almost sticks symmetry with rows 78 swapped:
Applying the symmetry, i.e. swap bands 23, colums 12,46,89, and renumber [(1)(2)(3)](45)(67)(89)
gives the same bands 23 and band 1 will change to

Code: Select all

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

Now swap r23:

Code: Select all

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

So we have a unique puzzle with a solution, which just differs in the 123 order in r23.
But we can swap the 123's in the 2 rows in a solution to get another one, which then is equal in all other cells - and therefore a solution to the original puzzle.
Since in row 1 c12,c46,and c89 are exchanged, they cannot be 123 (must be a pair out of 45,67,89), because then the transformation would lead to another solution of the puzzle.
So r1c3=3,r1c5=1,r1c7=2, which solves the puzzle (additionally bands 23 must fit to the sticks symmetry).

Thanks for the puzzles.

[Edit: corrected typos]

[eleven]

Another example for the 123+ symmetry.

Code: Select all

...123..7371......8..5......3.7.8.4...8...9...5.6.9.3....4....9......1636..231... (ER 9.9)

Code: Select all

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

Apart from r19c456 it has a symmetry half-turn (180° rotation, mirror at the center), swap columns 46, renumber 1-9 to 123547698
(or mirror b147/b963 at the center and b258 horizontally, and renumber):

Code: Select all

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

Now we can swap r1/9c456, because that does not effect the rest (in the columns the same 2 givens, in rows/boxes the same 3 are seen) - to get the original puzzle.
The solution must have this symmetry as well, otherwise applying it would give another solution to the same puzzle.
By this transformation cells r5c456 are mapped to themselves, so they can only be 123 (have to be same by renumbering),
and the digit symmetry must hold for all cells (apart from r19c456).

Code: Select all

+-------------------+-------------------+-------------------+
| 459   469   4569  |  1     2     3    | 468   89    7     |
| 3     7     1     | #89   #89+6  46   | 2456  259   2456  |
| 8     2469  2469  |  5     679   467  | 3     1     46    |
+-------------------+-------------------+-------------------+
| 129   3     269   |  7     5     8    | 26    4     126   |
| 47    46    8     |  3     1     2    | 9     57    56    |
| 127   5     27    |  6     4     9    | 278   3     128   |
+-------------------+-------------------+-------------------+
| 57    1     3     |  4     678   567  | 2578  2578  9     |
| 2457  248   2457  | #89   #89+7  57   | 1     6     3     |
| 6     89    579   |  2     3     1    | 4578  578   458   |
+-------------------+-------------------+-------------------+

The externals of the UR 89, 6r2c5 and 7r8c5 are symmetric, so both must be true.
Solves e.g. with a kite 6.

[yzfwsf]

Another example for the 123+ symmetry.

Code: Select all

...123..7371......8..5......3.7.8.4...8...9...5.6.9.3....4....9......1636..231... (ER 9.9)

Actually the solution to this puzzle has Sticks Symmetry.

Code: Select all

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

[AnotherLife]

Hi YZF,
You are not quite right. This is the solution of Eleven's puzzle:

Code: Select all

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

If we make the following permutation of rows/columns 456123978/312645789 then we will get your matrix. But how can it help us to solve the puzzle?

[AnotherLife]

YZF, as you don't answer my question, let me make it more clear. Your program offers us an alternative way to solve Eleven's puzzle by considering Sticks Symmetry (not '123+ symmetry' introduced my Marek). Would you please explain us that solution in detail? I think it may be interesting not only to me.

[yzfwsf]

Unlike human solvers, programs detect symmetries by checking whether the solution satisfies the symmetry, not the clues.

[AnotherLife]

As far as I see, your solver uses Brute Force to solve this puzzle and then checks whether the solution has some symmetry. Indeed, if we make the following permutation of rows/colums 978456231/213546798 and digits 321674589, this matrix will not change.

Code: Select all

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

But this transposition turns some givens into values and vice versa, so our initial puzzle is changed. On the other hand, your solver presents Gurth's symmetry placement as a method that enables us to solve this puzzle. What conclusion based on Sticks symmetry (not '123+ symmetry') can we make when we face only the givens?

[yzfwsf]

If you only test the clues, of course there is no Sticks symmetry, but it also needs to be sure that the puzzle has a unique solution, so in general there is a difference between the program and the human solver.The program must call the brute force solver.

[AnotherLife]

The program must call the brute force solver.

I am glad that we agree at least about this point. But that is not a solution:

Code: Select all

Gurth's symmetry placement: Need rearrange rows to 456123978 and rearrange cols to 312645789 => r4c3<>6,r4c7<>56,r6c3<>47,r6c7<>7
Candidate's mapping in Sticks Symmetry Type 1: 1<=>3 2<=>2 4<=>6 5<=>7 8<=>8 9<=>9

It is not justified by means of logic. Sorry.

P.S. Eleven, what do you think about it?

[eleven]

I found it interesting, that there is a "normal" sticks symmetry in the solution.
Since it cannot be shown by the givens, guessing it to solve the puzzle is like guessing backdoors of a puzzle.