The New Sudoku Players' Forum

[eleven]

of a bunch of puzzles, i tried last week:

Code: Select all

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

[denis_berthier]

.

Code: Select all

Resolution state after Singles and whips[1]:
   +-------------------+-------------------+-------------------+
   ! 2479  6     249   ! 79    3     8     ! 5     279   1     !
   ! 3     279   1     ! 479   247   5     ! 8     6     279   !
   ! 279   5     8     ! 1     6     29    ! 279   4     3     !
   +-------------------+-------------------+-------------------+
   ! 6     29    2459  ! 3     458   1     ! 279   279   2789  !
   ! 8     1     7     ! 2     9     6     ! 3     5     4     !
   ! 2459  239   23459 ! 458   458   7     ! 6     1     289   !
   +-------------------+-------------------+-------------------+
   ! 1     4     2359  ! 5789  2578  239   ! 279   279   6     !
   ! 2579  8     259   ! 6     257   4     ! 1     3     2579  !
   ! 2579  2379  6     ! 579   1     239   ! 4     8     2579  !
   +-------------------+-------------------+-------------------+
125 candidates.

There's a simplest-first solution in W4:

Code: Select all

hidden-pairs-in-a-row: r4{n4 n5}{c3 c5} ==> r4c5≠8, r4c3≠9, r4c3≠2
hidden-single-in-a-row ==> r4c9=8
finned-x-wing-in-rows: n5{r4 r7}{c3 c5} ==> r8c5≠5
finned-x-wing-in-columns: n7{c2 c5}{r2 r9} ==> r9c4≠7
z-chain[3]: r9c4{n9 n5} - c9n5{r9 r8} - r8n9{c9 .} ==> r9c1≠9
z-chain[3]: r9c4{n9 n5} - c9n5{r9 r8} - r8n9{c9 .} ==> r9c2≠9
t-whip[3]: r9c4{n9 n5} - r7n5{c5 c3} - r7n3{c3 .} ==> r7c6≠9
biv-chain[4]: r3n7{c7 c1} - c2n7{r2 r9} - r9n3{c2 c6} - c6n9{r9 r3} ==> r3c7≠9
biv-chain[3]: r3c7{n7 n2} - r3c6{n2 n9} - r1c4{n9 n7} ==> r1c8≠7
biv-chain[3]: b3n9{r1c8 r2c9} - b3n7{r2c9 r3c7} - b6n7{r4c7 r4c8} ==> r4c8≠9
biv-chain[3]: r1n7{c4 c1} - c1n4{r1 r6} - c4n4{r6 r2} ==> r2c4≠7
biv-chain[4]: b2n7{r1c4 r2c5} - c2n7{r2 r9} - r9n3{c2 c6} - c6n9{r9 r3} ==> r1c4≠9
naked-single ==> r1c4=7
biv-chain[3]: b1n7{r2c2 r3c1} - r3c7{n7 n2} - b2n2{r3c6 r2c5} ==> r2c2≠2
whip[2]: r2n2{c5 c9} - b9n2{r8c9 .} ==> r7c5≠2
biv-chain[3]: b3n7{r3c7 r2c9} - r2c2{n7 n9} - r4n9{c2 c7} ==> r4c7≠7
hidden-single-in-a-block ==> r4c8=7
biv-chain[3]: c8n2{r7 r1} - b3n9{r1c8 r2c9} - r6c9{n9 n2} ==> r8c9≠2, r9c9≠2
whip[1]: b9n2{r7c8 .} ==> r7c3≠2, r7c6≠2
stte

There's no 1- or 2- step solution with whips of any reasonable length (here reasonable would be max 6, but I tested up to 8).

Using the fewer-steps algorithm (http://forum.enjoysudoku.com/reducing-the-number-of-steps-t39234.html), I found a solution in W6 in 4 non-W1 steps (instead of the 16 above in W4):

Code: Select all

=====> STEP #1
hidden-pairs-in-a-row: r4{n4 n5}{c3 c5} ==> r4c3≠2, r4c5≠8, r4c3≠9
hidden-single-in-a-row ==> r4c9=8

=====> STEP #2
whip[6]: c9n5{r8 r9} - c9n7{r9 r2} - c2n7{r2 r9} - r9c4{n7 n9} - r7n9{c6 c3} - b7n3{r7c3 .} ==> r8c9≠9
whip[1]: r8n9{c3 .} ==> r7c3≠9, r9c1≠9, r9c2≠9

=====> STEP #3
whip[6]: c9n5{r9 r8} - c9n7{r8 r2} - c2n7{r2 r9} - r9c4{n7 n5} - r7n5{c5 c3} - b7n3{r7c3 .} ==> r9c9≠9
whip[1]: r9n9{c6 .} ==> r7c4≠9, r7c6≠9

=====> STEP #4
whip[6]: r9n3{c2 c6} - r7c6{n3 n2} - c5n2{r8 r2} - r2c2{n2 n9} - c9n9{r2 r6} - b4n9{r6c1 .} ==> r9c2≠7

stte

This is the solution obtained at the 1st try and 9 more tries didn't give fewer steps.

[shye]

after an important naked triple in r4:

Code: Select all

.-------------------.-----------------.----------------.
| 2479  6     249   | 79    3     8   | 5    279  1    |
| 3    y279   1     |*479  *247   5   | 8    6   y279  |
| 279   5     8     | 1     6    x29  | 279  4    3    |
:-------------------+-----------------+----------------:
| 6    x29    45    | 3     45    1   | 279  279  8    |
| 8     1     7     | 2     9     6   | 3    5    4    |
| 2459  239   23459 | 458   458   7   | 6    1   x29   |
:-------------------+-----------------+----------------:
| 1     4     2359  | 5789  2578  3-29|x279 x279  6    |
| 2579  8     259   | 6     257   4   | 1    3   *2579 |
| 2579  2379  6     | 579   1     239 | 4    8   *2579 |
'-------------------'-----------------'----------------'     
                                                              (xy: 29)
                                                           => -xr7c6 stte
             yr2c2 - (y=x)r4c2 - xr4c78 = xr6c9
             //                                \
(x=y)r3c6 - yr2c45                            xr89c9 = xr7c78
             \\                                /
             yr2c9 - (y=x)r6c9----------------


very remote pair fun puzzle!

edit: updated my notation (thanks jco!) it was a bit scuffed in places

[marek stefanik]

Code: Select all

.-------------------.-----------------.----------------.
| 2479  6     249   | 7–9   3     8   | 5    279  1    |
| 3     279   1     |a479  a247   5   | 8    6    279  |
| 279   5     8     | 1     6    b29  | 279  4    3    |
:-------------------+-----------------+----------------:
| 6    a29    45    | 3     45    1   | 279  279  8    |
| 8     1     7     | 2     9     6   | 3    5    4    |
| 2459  239   23459 | 458   458   7   | 6    1   a29   |
:-------------------+-----------------+----------------:
| 1     4     2359  | 5789  2578  3–29|a279 a279  6    |
| 2579  8     259   | 6     257   4   | 1    3    2579 |
| 2579  2379  6     | 579   1     239 | 4    8    2579 |
'-------------------'-----------------'----------------'

Whichever digit appears in r4c2 is forced into c9 in r6, b2 in r2, where it creates a pair with r3c6, and r7 in b9.
Since we know it's not the same digit as in r3c6, we can also eliminate 29 from r7c6. stte

Xsudo input: Show

8 Truths = {29R2 29R6 29B9 3N6 4N2}
14 Links = {29r7 29c2 29c6 29c9 29b2 29b4 29b6}

Multi-links: Show

It's simple to prove the elimination of 9r1c4.
r4c2 forces the finned x-wing (r26\c29b4 + extra fins) to take a fin in b2.
We can cover the x-wing like this (the 1s and 0s will be helpful later, for now just ignore them):

Code: Select all

.---------.---------.---------.
| .  .  . | .  .  . | .  .  . |
| .  2  . | 1  1  . | .  .  2 |
| .  .  . | .  .  0 | .  .  . |
:---------+---------+---------:
| .  2  . | .  .  . | .  .  . |
| .  .  . | .  .  . | .  .  . |
| 2  3  2 | .  .  . | .  .  2 |
:---------+---------+---------:
| .  .  . | .  .  0 | 0  0  . |
| .  .  . | .  .  . | .  .  1 |
| .  .  . | .  .  . | .  .  1 |
'---------'---------'---------'

(2L)A = (5L)r26c29b4 / 2
Again, it doesn't matter that r6c2 appears in three of the links, it still leaves enough for another true candidate.
6 truths: 29r2, 29r6, r3c6, r4c2
6 links: (2L)2A, (2L)9A, 29b2
We get a rank0 area in b2 => –9r1c4
Xsudo input for this partial pattern:
6 Truths = {29R2 29R6 3N6 4N2}
8 Links = {29c2 29c9 29b2 29b4}

The other two eliminations are a bit tricky, since if we want to obtain them at once, we need to apply a more complicated fish link.
If we just add r7c6b29 on the previous diagram, we get 9 links in total, covering every cell twice.
Therefore we get a 4-link.
If we now replace the links in our original Xsudo input with 29 in this 4-link, we reduce the pattern to rank0.
29r7c6 are now covered by the links, but not part of the truths, so they can be eliminated.

Marek

[DEFISE]

.
... I found a solution in W6 in 4 non-W1 steps (instead of the 16 above in W4).

This time I have not found better.

[denis_berthier]

.
... I found a solution in W6 in 4 non-W1 steps (instead of the 16 above in W4).

This time I have not found better.

How many tries did you do? I was lazy to go beyond 10.

[DEFISE]

Hi Denis,

I did only 20 tries, but after reading shye's solution, I realize that there is a solution in gW5 with 3 steps, that my algo couldn't find.
As I don't have time today to write the g-whips with the correct syntax, I summarize this solution has follows:

16 singles
Block/Line : 4r2b2 => -4r1c4
Hidden pairs: 45r4c35 => -2r4c3 -9r4c3 -8r4c5
Single: 8r4c9
g-whip[5] to eliminate 2r7c6
g-whip[5] to eliminate 9r7c6
STTE

N.B: 2r7c6 and 9r7c6 cannot be eliminated by any whip and even by any braid.

[totuan]

Hi marek & shye, nice path!

Code: Select all

 *--------------------------------------------------------------------*
 | 2479   6      249    | 79     3      8      | 5      279    1      |
 | 3      279    1      | 479    247    5      | 8      6      279    |
 | 279    5      8      | 1      6      29     | 279    4      3      |
 |----------------------+----------------------+----------------------|
 | 6      29     4-5    | 3     #45     1      | 279    279    8      |
 | 8      1      7      | 2      9      6      | 3      5      4      |
 | 2459   239    23459  |*458   *458    7      | 6      1      29     |
 |----------------------+----------------------+----------------------|
 | 1      4     #2359   |*5789  *2578   239    | 279    279    6      |
 | 2579   8      259    | 6      257    4      | 1      3      2579   |
 | 2579   2379   6      | 579    1      239    | 4      8      2579   |
 *--------------------------------------------------------------------*

My path for this one – series of URs
01: UR(58)r67c45: (5)r4c5=(5)r7c3 => r4c3<>5, some singles

Code: Select all

 *-----------------------------------------------------------*
 | 4     6     29    | 79    3     8     | 5     279   1     |
 | 3    #279   1     |*49-7 *24-7  5     | 8     6    #279   |
 | 279   5     8     | 1     6     29    | 279   4     3     |
 |-------------------+-------------------+-------------------|
 | 6     29    4     | 3     5     1     | 279   279   8     |
 | 8     1     7     | 2     9     6     | 3     5     4     |
 | 259   239   2359  |*48   *48    7     | 6     1     29    |
 |-------------------+-------------------+-------------------|
 | 1     4     2359  |*5789 *278   239   |#279  #279   6     |
 | 2579  8     259   | 6     27    4     | 1     3     2579  |
 | 2579  2379  6     | 579   1     239   | 4     8     2579  |
 *-----------------------------------------------------------*

02: MUG(478)r267c45: (7)r2c29=r7c78-r89c9=r2c9 => r2c45<>7, r1c4=7

Code: Select all

 *-----------------------------------------------------------*
 | 4     6     29    | 7     3     8     | 5     29    1     |
 | 3     279   1     | 49    24    5     | 8     6     279   |
 | 279   5     8     | 1     6     29    | 279   4     3     |
 |-------------------+-------------------+-------------------|
 | 6     29    4     | 3     5     1     |*279  *279   8     |
 | 8     1     7     | 2     9     6     | 3     5     4     |
 | 259   239   2359  | 48    48    7     | 6     1     29    |
 |-------------------+-------------------+-------------------|
 | 1     4     2359  | 589   278   39-2  |*279  *279   6     |
 | 2579  8     259   | 6     27    4     | 1     3     2579  |
 | 2579  2379  6     | 59    1     239   | 4     8     2579  |
 *-----------------------------------------------------------*

03: UR(79)r47c78: (2)r7c78=r4c78-(2=9)r4c2/r6c9-r2c29=r2c4-(9=2)r3c6 => r7c6<>2
04: (3=9)r7c6-(9=5)r9c4-r7c4=r7c3 => r7c3<>3, stte

Thanks for nice puzzle!
totuan

[denis_berthier]

Hidden pairs: 45r4c35 => -2r4c3 -9r4c3 -8r4c5
Single: 8r4c9
g-whip[5] to eliminate 2r7c6
g-whip[5] to eliminate 9r7c6

Right, I reconstructed the resolution path from your 3 eliminations (any of the 3 will do for the Pairs). It's much faster to check a path than to find it.

Code: Select all

hidden-pairs-in-a-row: r4{n4 n5}{c3 c5} ==> r4c3≠2, r4c5≠8, r4c3≠9
hidden-single-in-a-row ==> r4c9=8
g-whip[5]: b9n2{r7c8 r789c9} - r6c9{n2 n9} - b4n9{r6c1 r4c2} - r2n9{c2 c4} - r3c6{n9 .} ==> r7c6≠2
g-whip[5]: b9n9{r7c8 r789c9} - r6c9{n9 n2} - b4n2{r6c1 r4c2} - r2n2{c2 c5} - r3c6{n2 .} ==> r7c6≠9
stte

a solution in gW5 with 3 steps, that my algo couldn't find..

That's the problem with this kind of steepest descent algorithm: the 1st g-whip[5] in your 3-step path has score 1, but there exists another g-whip[5] available instead of it, with score 8 and it's the only candidate with score 8 in gW5. It leads to a 4-step solution in gW5, in which your 2nd step re-appears, but as the 3rd one.

Code: Select all

=====> STEP #1
hidden-pairs-in-a-row: r4{n4 n5}{c3 c5} ==> r4c3≠2, r4c5≠8, r4c3≠9
hidden-single-in-a-row ==> r4c9=8

=====> STEP #2
g-whip[5]: b2n7{r1c4 r2c456} - c2n7{r2 r9} - c9n7{r9 r8} - c9n5{r8 r9} - r9c4{n5 .} ==> r1c4≠9
naked-single ==> r1c4=7

=====> STEP #3
g-whip[5]: b9n2{r7c8 r789c9} - r6c9{n2 n9} - r4n9{c8 c2} - r2n9{c2 c4} - r3c6{n9 .} ==> r7c6≠2

=====> STEP #4
whip[4]: c3n3{r6 r7} - r7c6{n3 n9} - r9c4{n9 n5} - r7n5{c4 .} ==> r6c2≠3
stte

[totuan]

Code: Select all

 *--------------------------------------------------------------------*
 | 2479   6      249    | 79     3      8      | 5      279    1      |
 | 3      279    1      | 479    247    5      | 8      6      279    |
 | 279    5      8      | 1      6      29     | 279    4      3      |
 |----------------------+----------------------+----------------------|
 | 6      29     45     | 3      45     1      | 279    279    8      |
 | 8      1      7      | 2      9      6      | 3      5      4      |
 | 2459   239    23459  | 458    458    7      | 6      1      29     |
 |----------------------+----------------------+----------------------|
 | 1      4      2359   | 5789   2578   239    | 279    279    6      |
 | 2579   8      259    | 6      257    4      | 1      3      2579   |
 | 2579   2379   6      | 579    1      239    | 4      8      2579   |
 *--------------------------------------------------------------------*

I studied more and see 3 steps
01: UR(79)r47c78: (2)r7c78=r4c78-(2=9)r4c2/r6c9-r2c29=r2c4-(9=2)r3c6 => r7c6<>2
02: (7)r9c2=r2c2-r2c5=r78c5 => r9c4<>7
03: (3=9)r7c6-(9=5)r9c4-r7c45=r7c3 => r7c3<>3, stte

Edit: oops – maybe one step: double UR(27|79)r47c78 with intenal 2’s and 9’s => r7c6<>29, stte
totuan

[eleven]

Same solution as shye and Marek, in my words:
The digit in r4c2 is also in r7c78, the other in r3c6 => -29r7c6, stte
xr4c2* - r4c78 = r6c9* - r89c9 = xr7c78
xr4c2 - *(r2c2|r2c9) = r2c45 - (x=y)r3c6

I also like totuan's way (as always), but can't see how the "double UR" could work.

[jco]

I also like totuan's way (as always), but can't see how the "double UR" could work.

Perhaps totuan has a better way, but the way I see, the double UR uses chains like the one's
in shye's solution, i.e.,

UR(27)r47c78 using internals => -9 r7c6
(9)r7c78 == (9)r4c78 - (9=2)r4c2 - r2c2 = r2c45 - (2=9)r3c6

UR(79)r7c78 using internals => -2 r7c6
(2)r7c78 == (2)r4c78 - r4c2 = r2c2 - r2c45 = (2)r3c6

Edit: corrected 2 typos.

[Cenoman]

Code: Select all

 +------------------------+-----------------------+----------------------+
 |  2479   6      249     |  79      3      8     |  5      279   1      |
 |  3     d279w   1       |  479v   C247    5     |  8      6   De279w   |
 |  279    5      8       |  1       6      29    |  279    4     3      |
 +------------------------+-----------------------+----------------------+
 |  6      29w    45      |  3       45     1     |  279w   279w  8      |
 |  8      1      7       |  2       9      6     |  3      5     4      |
 |  2459   239    23459   |  458     458    7     |  6      1     29x    |
 +------------------------+-----------------------+----------------------+
 |  1      4     c359-2   | b578-9 Bb578-2  3-29  |Fg279z Fg279z  6      |
 |  2579   8      259     |  6      B257    4     |  1      3   Ef2579y  |
 |  2579  c2379   6       |Aa579u    1      239   |  4      8   Ef2579y  |
 +------------------------+-----------------------+----------------------+

Kraken cell (579)r9c4
(5)r9c4 - r7c45 = (53-7)b7p38 = r2c2 - r2c9 = r89c9 - (7=29)r7c78
(7)r9c4 - r78c5 = r2c5 - r2c9 = r89c9 - (7=29)r7c78
(9)r9c4* - r2c4 = [r2c9 = r2c2 - r4c2 = r4c78] - (9=2)r6c9 - r89c9 = (2)r7c78
--------------------
=> -9 r7c46*, -2r7c356; ste

The main goal of this message is not to show a solution far from the puzzle maker expectations, but to say thank you eleven and shye for your interesting puzzles in these last days ! Thank you also to other puzzle posters !

[totuan]

I also like totuan's way (as always), but can't see how the "double UR" could work.

Code: Select all

 *--------------------------------------------------------------------*
 | 2479   6      249    | 79     3      8      | 5      279    1      |
 | 3      279    1      | 479    247    5      | 8      6      279    |
 | 279    5      8      | 1      6      29     | 279    4      3      |
 |----------------------+----------------------+----------------------|
 | 6      29     45     | 3      45     1      |*279   *279    8      |
 | 8      1      7      | 2      9      6      | 3      5      4      |
 | 2459   239    23459  | 458    458    7      | 6      1      29     |
 |----------------------+----------------------+----------------------|
 | 1      4      2359   | 5789   2578   239    |*279   *279    6      |
 | 2579   8      259    | 6      257    4      | 1      3      2579   |
 | 2579   2379   6      | 579    1      239    | 4      8      2579   |
 *--------------------------------------------------------------------*

Thank you and for separating:
UR(79)r47c78: (2)r7c78=r4c78-(2=9)r4c2/r6c9-r2c29=r2c4-(9=2)r3c6 => r7c6<>2
UR(27)r47c78: (9)r7c78=r4c78-(9=2)r4c2/r6c9-r2c29=r2c5-(2=9)r3c6 => r7c6<>9
I suggest to combine in one – not sure for the notation and syntax
UR(27|79)r47c78: (9|2)r7c78=(9|2)r4c78-(9|2=2|9)r4c2/r6c9-(2|9)r2c29=r2c5/r2c4-(2|9=9|2)r3c6 => r7c6<>9 & 2
Some one can count as 2 steps, but does not matter

totuan