The New Sudoku Players' Forum

[ixsetf]

This puzzle has a kind of symmetry which I personally haven't seen discussed on this forum.

Hidden Text: Show

Code: Select all

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


Play this puzzle online.

Below is the minimized version, it is much harder to do normally, but if my hunch is correct then it should be solvable by hand.

Code: Select all

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


Play this puzzle online.

[JC Van Hay]

The minimal one :

9r7c2 -> contradiction using basics; r1c2=9 and 44 solved cells.
3r8c6 -> contradiction using singles; r8c8=3; stte.

The easy one : 3 singles, then ...

Code: Select all

+--------------------+----------------------+-------------------+
| 1    36(9)   358   | 4      23(9)  238    | 7      56    256  |
| 369  2       347   | 3(19)  5      37(1)  | 346    8     46   |
| 358  347     34578 | 238    237    6      | 2345   1     9    |
+--------------------+----------------------+-------------------+
| 7    346     2345  | 23589  2349   23458  | 1      4569  4568 |
| 35   8       1345  | 1359   6      13457  | 459    2     457  |
| 256  146     9     | 1258   1247   124578 | 4568   4567  3    |
+--------------------+----------------------+-------------------+
| 4    3-1(9)  1238  | 7      123    1235   | 25689  569   2568 |
| 29   5       27(1) | 6      8      24(1)  | 249    3     247  |
| 238  37      6     | 235    234    9      | 2458   457   1    |
+--------------------+----------------------+-------------------+
[9r71c2 9r1c5,9r2c4 1r2c46 1r8c63] - 1r7c2; stte

[Leren]

Long chain single step solution for the easy one:

Code: Select all

*--------------------------------------------------------------------------------*
| 1       69      358      | 4       239     238      | 7       56      256      |
| 69      2       347      | 139     5       137      | 346     8       46       |
| 358    p347     34578    | 238    q237     6        | 2345    1       9        |
|--------------------------+--------------------------+--------------------------|
| 7       346     2345     | 23589   2349    23458    | 1      g4569    4568     |
| 35      8      b1345b    | 1359    6      j13457    |h459     2      k457      |
| 256   ua1-46a   9        | 1258   r1247    124578   | 4568  lg4567    3        |
|--------------------------+--------------------------+--------------------------|
| 4      t139     1238     | 7      s123     1235     | 25689   569     2568     |
| 29      5      c127      | 6       8      d124      | 249     3       247      |
| 238    n37      6        | 235    e234     9        | 2458  mf457     1        |
*--------------------------------------------------------------------------------*

- 1 r6c2                                        =  r5c3 - 4 r5c3\
                                                                 |
- 1 r6c2 = r5c3 - r8c3 = (1-4) r8c6 = r9c5 - r9c8 = r46c8 - r5c7 = r5c6 - 7 r5c6 \
                                                                                 |
  1 r6c2 = r7c2 - r7c5 = (1-7) r6c5 = r3c5 - r3c2 = r9c2 - r9c8 = r6c8  - 7 r5c9 /

=> 46r6c2; stte

The solution has symmetry in that boxes 2 and 7 are the same and boxes 3 and 4 are the same. Is this provable without solving the puzzle ? I doubt it.

Leren

[blue]

The solution has symmetry in that boxes 2 and 7 are the same and boxes 3 and 4 are the same. Is this provable without solving the puzzle ? I doubt it.

Right. The (unique) solution grid has 6 automorphisms, but none of the corresponding puzzles match.
The puzzle itself, doesn't have a non-trivial automorphism.
Without that, I don't think you can prove anything.

Code: Select all

1..4..7...2..5..8......6..97.....1...8..6..2...9.....34..7......5..8..3...6..9..1
1..4..7......5..8...3..6..97..5..1...8.....2...9.....34..7..2...5..8..3...6..9...
...4..7...2..5..8...3..6..97.....1...8.....2...9..4..34..7..2...5..8......6..9..1
..5..8..2...9..3...4..7..1...2..3..83.....9...1.....7...8..5..69..6..4...7..4....
.....8..26..9..3...4..7..1...2.....83.....9...1..2..7...8..5..69..6......7..4..5.
..5..8..26..9..3......7..1...2.....83..1..9...1.....7...8..5...9..6..4...7..4..5.


The puzzle would have 3 automorphisms, if the rest of the diagonal was filled in.
It's a singles puzzle, though.

[ixsetf]

Allow me to describe the kind of symmetry this puzzle has.

Consider the typical numbering of cells within a box.

Hidden Text: Show

Code: Select all

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


If you take the set of all cells with the same number, you a set like the following.

Hidden Text: Show

Code: Select all

+-------+-------+-------+
| X . . | X . . | X . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| X . . | X . . | X . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| X . . | X . . | X . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+


Moving any of these sets on the mini-diagonal towards the bottom right (wrapping within each box) yields 1=>2, 2=>3, 3=>1, 4=>5, 5=>6, 6=>4, 7=>8, 8=>9, 9=>7. Note how elements of each of the following sets stay within themselves, {1, 2, 3}, {4, 5, 6}, {7, 8, 9}. If my hunch is correct than the fact that it cant be shown false from the givens implies that it must be true for the completed puzzle. I haven't yet found a counterexample and I currently have a vague idea of how I would go about proving it.

Others have noted that boxes 2 and 7, as well as 3 and 4 are the same. This symmetry and the fact that cell 1 in each box is the same is sufficient to show this.

[blue]

If my hunch is correct than the fact that it cant be shown false from the givens implies that it must be true for the completed puzzle. I haven't yet found a counterexample and I currently have a vague idea of how I would go about proving it.

Would you consider this to be a counterexample ?

Code: Select all

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

[ixsetf]

If my hunch is correct than the fact that it cant be shown false from the givens implies that it must be true for the completed puzzle. I haven't yet found a counterexample and I currently have a vague idea of how I would go about proving it.

Would you consider this to be a counterexample ?

Code: Select all

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

Yes, that is indeed a counterexample, I should have checked more thoroughly before making claims! I had only checked for counterexamples that were subsets of the following pattern.

Hidden Text: Show

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 . |
| . . x | . . x | . . x |
+-------+-------+-------+


I will have to look into whether more restrictions would make statements about the whole puzzle possible. The property most likely will only hold for subsets of the above pattern.

Well either way, at least my above post describes the sort of symmetry that my puzzle contains.

[eleven]

This puzzle has a kind of symmetry which I personally haven't seen discussed on this forum.

The symmetries are described and discussed in this thread, with lists of the 26 basic and 122 possible digital symmetries.
For the mini-diagonal symmetry you can find examples here.

[blue]

I had only checked for counterexamples that were subsets of the following pattern.

Hidden Text: Show

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 . |
| . . x | . . x | . . x |
+-------+-------+-------+


I will have to look into whether more restrictions would make statements about the whole puzzle possible. The property most likely will only hold for subsets of the above pattern.

I wondered about that.
Here are some better counterexamples.

Code: Select all

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

Code: Select all

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

[ixsetf]

Indeed it does look like my proposition was false, it would be interesting if there was some criteria for which the property held, but it seems it will be quite difficult to find.