## Opposites of whatever digit is found in R5C5

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Opposites of whatever digit is found in R5C5

In the solution grid:
Let k be the digit present in R5C5
The cells opposite to the cells containing that digit k, contain each digit uniquely

(Note that the opposite cell of RiCj is R(10-i)C(10-j).)

An easy puzzle:
Code: Select all
..2|.5.|4..
9..|6..|1..
8.4|27.|...
---+---+---
...|..6|.85
...|...|9..
.7.|4..|.6.
---+---+---
..1|.2.|...
...|...|3.9
.6.|...|...

More difficult and possibly chains:
Code: Select all
258|...|..9
...|...|.7.
7..|...|...
---+---+---
.2.|..9|...
...|6.4|1..
8..|.5.|...
---+---+---
..9|53.|6..
...|...|.8.
1..|.98|...
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Opposites of whatever digit is found in R5C5

Unless I have misunderstood your extra constraint, neither puzzle is valid.

The first has no solution. The candidates for r5c5 are 1, 3, 8. Looking at each, there is a 1 in r2c7 but also a one in r7c3 so a one cant go in r8c3. Same for 8. There is an 8 in r4c8 but 46c2 has a 7 so that leaves k = 3.
Two paths show 0 solutions. Either solve via normal methods and find a naked 3 in r4c3, but there is already a 2 in r6c7, or place a 3 in r2c3 (opposite of the given 3 in r8c7), the naked 5 in r2c2 and be left with only possibles for r13c2 as 1.

In the second, the candidates for r5c5 are 2, 7 and 8. Can't be 8 by the extra constraint. Try 2 and there is no solution. Try 7 and there are multiple solutions...
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Opposites of whatever digit is found in R5C5

The unique solution for the first puzzle:

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

A solution for the second puzzle:

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

Haven't confirmed if the second puzzle has a unique soluton yet. If anyone claims it has multiple solutions, an alternative solution must be provided as proof. Otherwise it would be deemed a groundless accusation.

It would be very appreciated if dyitto could send me a brief demonstration of how to prove the uniqueness for the second puzzle via private message. I don't think I have enough grit to try solving it logically.

simon_blow_snow

Posts: 85
Joined: 26 December 2010

### Re: Opposites of whatever digit is found in R5C5

Hi Simon, both your solutions are correct.

I didn't try any manual solving (If only I'd had more time...), I'm only relying on my program.
It concludes uniqueness after a full backtrack/depth-first search - which is not regarded as elegant solving technique.
Also - with the regular solving techniques as far as I've implemented - it reaches this stage:

Code: Select all
*-----------------------------------------------------------------------------------*
|       2        5        8 |    1347     1467     1367 |      34     1346        9 |
|     369    13469     1346 |  123489    12468    12356 |   23458        7  1234568 |
|       7    13469     1346 |  123489    12468    12356 |   23458   123456  1234568 |
|---------------------------+---------------------------+---------------------------|
|     356        2   134567 |    1378      178        9 |   34578     3456   345678 |
|     359      379      357 |       6      278        4 |       1     2359    23578 |
|       8   134679    13467 |    1237        5     1237 |    2347    23469    23467 |
|---------------------------+---------------------------+---------------------------|
|       4        8        9 |       5        3      127 |       6       12      127 |
|      35       37     2357 |    1247    12467     1267 |       9        8   123457 |
|       1      367    23567 |     247        9        8 |   23457     2345    23457 |
*-----------------------------------------------------------------------------------*
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dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Opposites of whatever digit is found in R5C5

dyitto wrote:The cells opposite to the cells containing that digit k, contain each digit uniquely

ah, i was wondering if i had interpreted that correctly.

Each of the cells opposite k contain a unique digit...
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Opposites of whatever digit is found in R5C5

dyitto

So I guess the second puzzle is not meant to be solved by human?

I did try to solve it via repeated trial and error (with the help of a solver software), and found the answer in the tenth branch or so, and then I felt too exhausted to continue on the remaining 20-30 branches manually. I don't think uniqueness of puzzles should be verified that way.

Are you still attempting the Mystery Symbols Sudoku? Hope my reply earlier didn't discourage you too much to continue working on it. You are actually halfway to the correct answer.

simon_blow_snow

Posts: 85
Joined: 26 December 2010

### Re: Opposites of whatever digit is found in R5C5

simon_blow_snow wrote:So I guess the second puzzle is not meant to be solved by human?

I can promise you the following is nice to be solved by human:
Code: Select all
1..|6..|7.2
...|...|...
7..|4..|5.3
---+---+---
4..|78.|6..
..2|...|149
.9.|...|8..
---+---+---
...|...|4..
...|...|...
3..|15.|...

simon_blow_snow wrote:Are you still attempting the Mystery Symbols Sudoku?

No. I'm beaten.
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Opposites of whatever digit is found in R5C5

dyitto wrote:In the solution grid:
Let k be the digit present in R5C5
The cells opposite to the cells containing that digit k, contain each digit uniquely

Hi dyitto,

Just to understand this from a programming point of view. This is a vanilla sudoku with 1extra 9-cells region. The cells of this region have an 180 degree symmetry relationship withthe 9-cells of the solution grid that have the symbol in R5C5. Is this description accurate ?

tarek

tarek

Posts: 3537
Joined: 05 January 2006

### Re: Opposites of whatever digit is found in R5C5

hi tarek,

tarek wrote:This is a vanilla sudoku with 1extra 9-cells region. The cells of this region have an 180 degree symmetry relationship with the 9-cells of the solution grid that have the symbol in R5C5. Is this description accurate ?

tarek

with the 8-cells of the solution grid that have the symbol in R5C5
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Opposites of whatever digit is found in R5C5

Tareks description is correct.
@BryanL R5C5 is part of the extra 9-cells region, since R5C5 maps to itself no matter how you rotate.
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Opposites of whatever digit is found in R5C5

@BryanL & @dyitto,
Thank you both.

@dyitto,
You can see the potential (obviously you have already) in exploring other symmetries as wel ... so you have the 180 rotation, orthogonal reflections & diagonal reflections. The thing you need to watch out for when making the puzzle is to make sure that the 9 cells of the extra region have 9 different symbols (in other words you need to make sure that the 9-cells of the solution drid that contain the R5C5 symbol do not map onto each other after applying the symmetry constraint, I also guess you know this already) ....

Step 1: choose an S2 symmetry.
Step 2: choose a suitable solution grid (you can reshuffle a solution grid until it is suitable)
Step 3: Remove redundant clues (preferably in a symmetrical fashion following that S2 symmetry constarint) until you get a valid minimal symmetrical puzzle

Well done dyitto,

tarek

tarek

Posts: 3537
Joined: 05 January 2006