The New Sudoku Players' Forum

[m_b_metcalf]

Occasionally, puzzles are posted here and in the Patterns Game where the values of the givens are symmetrical. Symmetry is normally understood to mean s(i, j) = 10-s(10-i, 10-j) (after renumbering when necessary).

I wondered whether puzzles exist where the symmetry means s(i, j) = s(10-i, 10-j). (Perhaps this has been discussed in the distant past, but I can't find any references.) The puzzle below is a near miss: without the added clue it would have six solutions.

Code: Select all

 . . . . . 6 . . 1
 6 . 1 8 . . . . 5
 . 5 . . . 3 . 4 9
 2 . . . . . 7 . 3
 . . . . . . . . .
 3[6]7 . . . . . 2
 9 4 . 3 . . . 5 .
 5 . . . . 8 1 . 6
 1 . . 6 . . . . .  hard

.....6..16.18....5.5...3.492.....7.3.........367.....294.3...5.5....81.61..6.....

No. of givens =  27, minimal, givens rotationally identical except for 6r6c2 (added to achieve uniqueness)

In the hope that this is of some interest (and see my post in the General Forum).

Regards,

Mike

[AnotherLife]

I wondered whether puzzles exist where the symmetry means s(i, j) = s(10-i, 10-j).

Hello, Mike,
If all the givens are rotationally identical then such a puzzle cannot have a unique solution.
Suppose there are two different values in two centrally symmetric cells. Let us substitute each value of the solution for the symmetric one, then the positions of the givens will not change, and we will get a different solution to the puzzle. Hence all the values must be rotationally identical. Then it comes to be that we have two identical values in column 5, which is a contradiction.

Code: Select all

.---------.---------.---------.
| a  b  c | d  *e f | g  h  i |
|         |         |         |
|         |         |         |
:---------+---------+---------:
|         |         |         |
|         |         |         |
|         |         |         |
:---------+---------+---------:
|         |         |         |
|         |         |         |
| i  h  g | f  *e d | c  b  a |
'---------'---------'---------'

Consequently, there is no unique solution to such a symmetric puzzle.

Regards, Bogdan

[m_b_metcalf]

Consequently, there is no unique solution to such a symmetric puzzle.

Quite. I'm therefore happy to have found at least one puzzle (see my first post in the General Forum) that has the minimum of only two solutions. Once you've fixed any unsolved cell at a correct value, the remainder of the puzzle has a unique solution. I'll try solving it by hand.

Regards,

Mike

[Edit: Solving by hand, I chose 7r5c4 as one of the two alternatives, and it becomes an SE 7.2 problem.]

[AnotherLife]

[Edit: Solving by hand, I chose 7r5c4 as one of the two alternatives, and it becomes an SE 7.2 problem.]

I wrote to you in the general forum that this variant is pretty good, and I think it is up to you to publish this puzzle. I am not sure it is possible to find a one-step solution to this puzzle.

[m_b_metcalf]

[Edit: Solving by hand, I chose 7r5c4 as one of the two alternatives, and it becomes an SE 7.2 problem.]

I wrote to you in the general forum that this variant is pretty good, and I think it is up to you to publish this puzzle. I am not sure it is possible to find a one-step solution to this puzzle.

As you wish:

Code: Select all

 . 6 1 8 . . 9 . 3
 8 . . 6 . . 2 4 1
 9 . 4 . . . . 8 .
 7 . 5 . . . . 9 .
 . . . [7] 5 . . . .
 . 9 . . . . 5 . 7
 . 8 . . . . 4 . 9
 1 4 2 . . 6 . . 8
 3 . 9 . . 8 1 6 .  Less hard than the first puzzle, but still not easy.

.618..9.38..6..2419.4....8.7.5....9....75.....9....5.7.8....4.9142..6..83.9..816.

The clue 7r5c4 added to reduce the number of solutions from two (isomorphic) to one.

Regards,

Mike

[AnotherLife]

The clue 7r5c4 added to reduce the number of solutions from two (isomorphic) to one.

Sorry, I meant this one. It is also rated 7.2 but much harder for a human. I added the clue 6r4c5.

Code: Select all

.----------------.-----------------------.----------------.
| 25   6     1   | 8       247    2457   | 9    57    3   |
| 8    357   37  | 6       379    3579   | 2    4     1   |
| 9    2357  4   | 12357   1237   12357  | 67   8     56  |
:----------------+-----------------------+----------------:
| 7    123   5   | 1234    6      1234   | 38   9     24  |
| 246  123   368 | 123479  5      123479 | 368  123   246 |
| 246  9     368 | 1234    12348  1234   | 5    123   7   |
:----------------+-----------------------+----------------:
| 56   8     67  | 12357   1237   12357  | 4    2357  9   |
| 1    4     2   | 3579    379    6      | 37   357   8   |
| 3    57    9   | 2457    247    8      | 1    6     25  |
'----------------'-----------------------'----------------'