The New Sudoku Players' Forum

[denis_berthier]

.
This is for those of you who like the really hard ones. (But it's still solvable without replacement or uniqueness.)

Code: Select all

+-------+-------+-------+
! . . 3 ! . 5 6 ! . . . !
! . . 7 ! 1 . 9 ! . 3 . !
! . 6 . ! 3 7 . ! . 5 . !
+-------+-------+-------+
! . 9 1 ! 5 6 . ! . . 7 !
! . 3 . ! 9 . 7 ! . . . !
! 7 . . ! . . . ! 5 . . !
+-------+-------+-------+
! . . 9 ! . . . ! 8 6 5 !
! . . . ! . . . ! 4 . 2 !
! . . . ! 6 . 5 ! . 1 . !
+-------+-------+-------+
..3.56.....71.9.3..6.37..5..9156...7.3.9.7...7.....5....9...865......4.2...6.5.1.;121;47151
SER = 11.7

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 12489  1248   3      ! 248    5      6      ! 1279   24789  1489   !
   ! 2458   2458   7      ! 1      248    9      ! 26     3      468    !
   ! 12489  6      248    ! 3      7      248    ! 129    5      1489   !
   +----------------------+----------------------+----------------------+
   ! 248    9      1      ! 5      6      2348   ! 23     248    7      !
   ! 24568  3      24568  ! 9      1248   7      ! 126    248    1468   !
   ! 7      248    2468   ! 248    12348  12348  ! 5      2489   134689 !
   +----------------------+----------------------+----------------------+
   ! 1234   1247   9      ! 247    1234   1234   ! 8      6      5      !
   ! 13568  1578   568    ! 78     1389   138    ! 4      79     2      !
   ! 248    2478   248    ! 6      2489   5      ! 379    1      39     !
   +----------------------+----------------------+----------------------+
186 candidates.

[DEFISE]

"Simplest-first" in W6 =>

Box/Line: 3b9r9 => -3r9c1 -3r9c5
whip[3]: r8c4{n8 n7}- r8c8{n7 n9}- r9n9{c7 .} => -8r9c5
Box/Line: 8r9b7 => -8r8c1 -8r8c2 -8r8c3
whip[3]: r5n5{c1 c3}- r8c3{n5 n6}- c1n6{r8 .} => -2r5c1
whip[3]: r5n5{c1 c3}- r8c3{n5 n6}- c1n6{r8 .} => -4r5c1
whip[3]: r5n5{c1 c3}- r8c3{n5 n6}- c1n6{r8 .} => -8r5c1
whip[4]: c8n4{r6 r1}- c4n4{r1 r7}- c4n7{r7 r8}- c8n7{r8 .} => -4r6c9
whip[6]: r6n3{c6 c9}- r9c9{n3 n9}- r8n9{c8 c5}- r8n3{c5 c1}- r8n6{c1 c3}- r6n6{c3 .} => -3r4c6
Single(s): 3r4c7, 3r9c9

Code: Select all

|-----------------------------------------------------------|
| 12489 1248  3     | 248   5     6     | 1279  24789 1489  |
| 2458  2458  7     | 1     248   9     | 26    3     468   |
| 12489 6     248   | 3     7     248   | 129   5     1489  |
|-----------------------------------------------------------|
| 248   9     1     | 5     6     248   | 3     248   7     |
| 56    3     24568 | 9     1248  7     | 126   248   1468  |
| 7     248   2468  | 248   12348 12348 | 5     2489  1689  |
|-----------------------------------------------------------|
| 1234  1247  9     | 247   1234  1234  | 8     6     5     |
| 1356  157   56    | 78    1389  138   | 4     79    2     |
| 248   2478  248   | 6     249   5     | 79    1     3     |
|-----------------------------------------------------------|

Tridagon (2,4,8) in b1p249, b2p159, b4p168, b5p357,
With 5 guardians: 1r1c2,5r2c1,5r5c3,6r5c3,1r5c5

The partial g-whip from 5r5c1 below proves that +5r5c1 => -1r1c2 and -5r2c1 :
c1n6{r5 r8}- c1n3{r8 r7}- c1n1{r7 r13}
So -5r5c3 => -1r1c2 and -5r2c1
So we can consider chains that see only the following 3 guardians: 5r5c3,6r5c3,1r5c5
Here is one :
Trid-OR3-ctr-g-braid[4]: r6n6{c9 c3}- r5c1{n6 n5}- r6n1{c9 c56}- OR3{{n5r5c3 n6r5c3 n1r5c5 |.}} => -9r6c9

Then the puzzle is solvable in W7 :
 

Hidden Text: Show

Single(s): 9r6c8, 7r8c8, 8r8c4, 9r9c7, 7r1c7, 9r8c5, 7r9c2, 7r7c4
whip[6]: r8c6{n3 n1}- r8c2{n1 n5}- r8c3{n5 n6}- r6n6{c3 c9}- r6n1{c9 c5}- r6n3{c5 .} => -3r7c6
whip[5]: r6n6{c9 c3}- r5c1{n6 n5}- r2n5{c1 c2}- c2n8{r2 r1}- c8n8{r1 .} => -8r6c9
whip[5]: r6n3{c5 c6}- r6n1{c6 c9}- r6n6{c9 c3}- r8n6{c3 c1}- r8n3{c1 .} => -8r6c5
whip[7]: r1c4{n4 n2}- r6c4{n2 n4}- c6n4{r6 r7}- c2n4{r7 r1}- r1c8{n4 n8}- c9n8{r1 r5}- c5n8{r5 .} => -4r2c5
whip[7]: r2c5{n8 n2}- r1c4{n2 n4}- r6c4{n4 n2}- c6n2{r6 r7}- c2n2{r7 r1}- r1c8{n2 n8}- c9n8{r1 .} => -8r5c5
Single(s): 8r2c5
whip[7]: r6c4{n4 n2}- b2n2{r1c4 r3c6}- r1c4{n2 n4}- c8n4{r1 r4}- r4n2{c8 c1}- c3n2{r5 r9}- r9c5{n2 .} => -4r5c5
whip[7]: r6c4{n2 n4}- b2n4{r1c4 r3c6}- r1c4{n4 n2}- c8n2{r1 r4}- r4n4{c8 c1}- c3n4{r5 r9}- r9c5{n4 .} => -2r5c5
Single(s): 1r5c5, 1r6c9, 6r6c3, 5r5c1, 5r8c3, 1r8c2, 3r8c6, 6r8c1, 1r1c1, 9r1c9, 5r2c2, 1r3c7, 9r3c1, 3r6c5, 1r7c6, 3r7c1
whip[3]: r2n2{c7 c1}- r3n2{c3 c6}- r4n2{c6 .} => -2r5c7
Single(s): 6r5c7, 2r2c7, 4r2c1, 6r2c9
whip[2]: r1n4{c4 c8}- r4n4{c8 .} => -4r3c6
STTE

[Cenoman]

Code: Select all

 +---------------------------+------------------------+--------------------------+
 |  12489    1248    3       |  248   5       6       |  1279   24789   1489     |
 |  2458     2458    7       |  1     248     9       |  26     3       468      |
 |  12489    6       248     |  3     7       248     |  129    5       1489     |
 +---------------------------+------------------------+--------------------------+
 |  248      9       1       |  5     6       2348    |  23     248     7        |
 |zv24568*   3      w2458-6* |  9     1248    7       |  126    248     1468     |
 |  7        248   yB2468    |  248   12348   12348   |  5      2489    134689   |
 +---------------------------+------------------------+--------------------------+
 |  1234     1247    9       |  247   1234    1234    |  8      6       5        |
 |  136-58* A157-8 xC56-8*   | a78    1389    138     |  4     b79      2        |
 |  248      2478    248     |  6    d249-8   5       | c379    1      c39       |
 +---------------------------+------------------------+--------------------------+

1. (8=7)r8c4 - (7=9)r8c8 - r9c79 = (9)r9c5 => -8 r9c5; (-8 r8c123)
2. UR(56)r58c13 using externals (5)r8c2 == (6)r6c3 - (6=5)r8c3 loop => -5 r8c1, -6 r5c3
3. (5)r5c1 = r5c3 - (5=6)r8c3 - r6c3 = (6)r5c1 => -248 r5c1

Code: Select all

 +------------------------+------------------------+---------------------------+
 |  12489  a1248*  3      |  248*  5       6       |  1279   24789    1489     |
 | A2458*   2458   7      |  1     248*    9       |  26     3        468      |
 |  12489   6      248*   |  3     7       248*    |  129    5        1489     |
 +------------------------+------------------------+---------------------------+
 |  248*    9      1      |  5     6      x2348*   | y23     248      7        |
 | B56      3      2458*  |  9    X1248*   7       | Y126    248     Y1468     |
 |  7       248* Ce2468   |  248*  12348   12348   |  5      2489 ZzDf136-489  |
 +------------------------+------------------------+---------------------------+
 | c1234   b1247   9      |  247   1234    1234    |  8      6        5        |
 | c1356   b157   d56     |  78    1389    138     |  4      79       2        |
 |  248     2478   248    |  6     249     5       |  379    1        39       |
 +------------------------+------------------------+---------------------------+

4. TH(248)b1245 (*) having five guardians. Note that 5r2c1 and 5r5c3 have the same parity (both conjugates of 5r5c1) => 4-kraken to be considered:
(1)r1c2 - r78c2 = (13-6)r78c1 = r8c3 - r6c3 = (6)r6c9
(5)r2c1 - (5=6)r5c1 - r6c3 = (6)r6c9
(3)r4c6 - r4c7 = (3)r6c9
(1)r5c5 - r5c79 = (1)r6c9
=> -489 r6c9; 7 placements

Code: Select all

 +------------------------+-----------------------+----------------------+
 |  12489   1248   3      |  24   5       6       |  7     248   1489    |
 |  2458    2458   7      |  1    248     9       |  26    3     468     |
 |  12489   6      248    |  3    7       248     |  129   5     1489    |
 +------------------------+-----------------------+----------------------+
 |  248     9      1      |  5    6      B248-3   | f23    248   7       |
 |  56      3      2458   |  9    1248    7       |  126   248   1468    |
 |  7      F248  Fd2468   | F24  A13-248 B12348   |  5     9    e136     |
 +------------------------+-----------------------+----------------------+
 |  1234    124    9      |  7    1234    1234    |  8     6     5       |
 |Db136     15   Ec56     |  8    9     Ca13      |  4     7     2       |
 |  248     7      248    |  6    24      5       |  39    1     39      |
 +------------------------+-----------------------+----------------------+

5. (3)r8c6 = (3-6)r8c1 = r8c3 - r6c3 = (6-3)r6c9 = (3)r4c8 => -3 r4c6; 1 placement
6. (3)r6c5 = r46c6 - r8c6 = (3-6)r8c1 = r8c3 - (6=248)r6c234 => -248 r6c5

Code: Select all

 +------------------------+----------------------+---------------------+
 |  12489  a1248   3      |zb24   5      6       |  7    b248  c1489   |
 |  2458   A2458   7      |  1   f8-24   9       |  26    3    c468    |
 |  12489   6      248    |  3    7      248     |  12    5    c1489   |
 +------------------------+----------------------+---------------------+
 |  248     9      1      |  5    6      248     |  3     248   7      |
 |  56      3      2458   |  9   d1248   7       |  126   248  d1468   |
 |  7      x248    2468   | y24   13     12348   |  5     9     16     |
 +------------------------+----------------------+---------------------+
 |  1234   X124    9      |  7   Z1234  Y1234    |  8     6     5      |
 |  1356    15     56     |  8    9      13      |  4     7     2      |
 |  248     7      248    |  6   Z24     5       |  9     1     3      |
 +------------------------+----------------------+---------------------+

7. Kraken column (2)r1267c2
(2)r1c2 - (24=8)r1c48 - r123c9 = r5c9 - r5c5 = (8)r2c5
(2)r2c2
(2)r6c2 - r6c4 = (2)r1c4
(2)r7c2 - r7c6 = (2)r79c5

8. Kraken column (4)r1267c2
(4)r1c2 - (24=8)r1c48 - r123c9 = r5c9 - r5c5 = (8)r2c5
(4)r2c2
(4)r6c2 - r6c4 = (4)r1c4
(4)r7c2 - r7c6 = (4)r79c5
=> -24 r2c5; 1 placement

Code: Select all

 +------------------------+----------------------+---------------------+
 |  12489   1248   3      | c24   5      6       |  7     248   1489   |
 |  245     245    7      |  1    8      9       |  26    3     46     |
 |  12489   6     a248    |  3    7     b24      |  12    5     1489   |
 +------------------------+----------------------+---------------------+
 |  248     9      1      |  5    6      248     |  3     248   7      |
 | B56      3     A2458   |  9   F1-24   7       |EC126   248 EC1468   |
 |  7       248   y2468   |zd24   13     12348   |  5     9    D16     |
 +------------------------+----------------------+---------------------+
 |  1234    124    9      |  7    1234   1234    |  8     6     5      |
 |  1356    15     56     |  8    9      13      |  4     7     2      |
 |  248     7     Y248    |  6   Z24     5       |  9     1     3      |
 +------------------------+----------------------+---------------------+

9 Kraken column (4)r3569c3
(4)r3c3 - (4=2)r3c6 - r1c4 = (2)r6c4
(4-5)r5c3 = (5-6)r5c1 = r5c79 - (6=1)r6c9 - r5c79 = (1)r5c5
(4)r6c3 - (4=2)r6c4
(4)r9c3 - (4=2)r9c5

10. Kraken column (2)r3569c3
(2)r3c3 - (2=4)r3c6 - r1c4 = (4)r6c4
(2-5)r5c3 = (5-6)r5c1 = r5c79 - (6=1)r6c9 - r5c79 = (1)r5c5
(2)r6c3 - (2=4)r6c4
(2)r9c3 - (2=4)r9c5
=> -24 r5c5; 16 placements

Code: Select all

 +--------------------+------------------+-------------------+
 |  1     248   3     |  24*  5    6     |  7    8-24  9     |
 |  24*   5     7     |  1    8    9     |  26*  3     46*   |
 |  9     6     248   |  3    7    24*   |  1    5     48    |
 +--------------------+------------------+-------------------+
 |  248*  9     1     |  5    6    248*  |  3    248#  7     |
 |  5     3     248   |  9    1    7     |  26   248   468   |
 |  7     248   6     |  24   3    248   |  5    9     1     |
 +--------------------+------------------+-------------------+
 |  3     24    9     |  7    24   1     |  8    6     5     |
 |  6     1     5     |  8    9    3     |  4    7     2     |
 |  248   7     248   |  6    24   5     |  9    1     3     |
 +--------------------+------------------+-------------------+

11. Almost RP (24)r2c79, r1c4 using external spoilers (2|4)r4c8
(2,4): [r2c79 = r2c1 - r4c1 *=* r4c6 - r3c6 = r1c4] = r4c8 => -24 r1c8; ste

[denis_berthier]

Code: Select all

|-----------------------------------------------------------|
| 12489 1248  3     | 248   5     6     | 1279  24789 1489  |
| 2458  2458  7     | 1     248   9     | 26    3     468   |
| 12489 6     248   | 3     7     248   | 129   5     1489  |
|-----------------------------------------------------------|
| 248   9     1     | 5     6     248   | 3     248   7     |
| 56    3     24568 | 9     1248  7     | 126   248   1468  |
| 7     248   2468  | 248   12348 12348 | 5     2489  1689  |
|-----------------------------------------------------------|
| 1234  1247  9     | 247   1234  1234  | 8     6     5     |
| 1356  157   56    | 78    1389  138   | 4     79    2     |
| 248   2478  248   | 6     249   5     | 79    1     3     |
|-----------------------------------------------------------|

Tridagon (2,4,8) in b1p249, b2p159, b4p168, b5p357,
With 5 guardians: 1r1c2,5r2c1,5r5c3,6r5c3,1r5c5

The partial g-whip from 5r5c1 below proves that +5r5c1 => -1r1c2 and -5r2c1 :
c1n6{r5 r8}- c1n3{r8 r7}- c1n1{r7 r13}
So -5r5c3 => -1r1c2 and -5r2c1
So we can consider chains that see only the following 3 guardians: 5r5c3,6r5c3,1r5c5
Here is one :
Trid-OR3-ctr-g-braid[4]: r6n6{c9 c3}- r5c1{n6 n5}- r6n1{c9 c56}- OR3{{n5r5c3 n6r5c3 n1r5c5 |.}} => -9r6c9

The problem with using ad hoc intermediate relations is, they don't change the resolution state. There's no Trid-OR3-ctr-g-braid[4] in this resolution state.
What there is is something I would call a bifurcating-partial-Trid-OR5-ctr-g-braid[4]:

Code: Select all

r6n6{c9 c3}- r5c1{n6 n5}- r6n1{c9 c56}- OR5{{n5r5c3 n6r5c3 n1r5c5 ||
                                                                  || n1r1c2 
                                                                  ||  n5r2c1
                                            }}

which you can probably complete into some bifurcating-Trid-OR5-g-braid[n] for some n >> 4, by adding two partial g-braids based on n1r1c2 and n5r2c1 that lead to their own contradictions.

OR-branching or bifurcation, whichever name you prefer, is something I've always avoided in all my chains, for many reasons:
- it amounts to adding one level of T&E,
- it introduces uncontrollable complexity (though, restricted to ORk-relations, this is less relevant),
- it's a form of reasoning by cases and therefore very inelegant.

[DEFISE]

Hi Denis,
I don't understand if you disagree with my logic or with my syntax.
So, considering the RS I got after my "Simplest first" in W6, could you tell me if you agree with the following 4 points:

1) There is an OR5 relation between the following 5 guardians: 1r1c2,5r2c1,5r5c3,6r5c3,1r5c5 i.e at least one of them must be true.

2) The partial g-whip c1n6{r5 r8}- c1n3{r8 r7}- c1n1{r7 r13} from 5r5c1
and (5r5c3 false => 5r5c1 true)
prove that: (5r5c3 false => 1r1c2 false and 5r2c1 false).

3) There is an OR3 relation between the 3 guardians 5r5c3,6r5c3,1r5c5 because if they were all false, then, according to (2), the 5 guardians of 1) would be false, which is impossible.

4) This partial g-braid r6n6{c9 c3}- r5c1{n6 n5}- r6n1{c9 c56} from 9r6c9 proves that :
9r6c9 true => 5r5c3,6r5c3,1r5c5 all false.
So, according to 3) 9r6c9 is false.

[denis_berthier]

Hi François,
I thought what I wrote was clear.

1) OK
2) ad hoc reasoning
3) improvement wrt to your first version (OR3-relation explicitly asserted)
4) worse than 1st version. If there's an OR3-relation, then there's an OR3-g-braid[4] concluding r6c9≠9. No need to recall the OR5 relation at this point.

As I said before, the problem is the ad hoc piece of reasoning in 2 leading to 3.
Where are the zillions of resolution rules that will allow this kind of reasoning in every possible case?
Said otherwise, where are the general rules allowing to pass from an OR5 relation to an OR3 one without having any candidate eliminated?

[DEFISE]

Ok I see.
Obviously I could have used this well-defined pattern to eliminate 9r6c9:
Trid-OR5-ctr-g-braid[7]: r6n6{c9 c3}- r5c1{n6 n5}- r6n1{c9 c56}- c1n6{r5 r8}- c1n3{r8 r7}- c1n1{r7 r13}- OR5{{all guardians |.}} => -9r6c9
François

[denis_berthier]

Ok I see.
Obviously I could have used this well-defined pattern to eliminate 9r6c9:
Trid-OR5-ctr-g-braid[7]: r6n6{c9 c3}- r5c1{n6 n5}- r6n1{c9 c56}- c1n6{r5 r8}- c1n3{r8 r7}- c1n1{r7 r13}- OR5{{all guardians |.}} => -9r6c9

Yes, it's a self-contained elimination and it is clearly pattern-based. You first solution was inference-based.
It may look computationally harder than your first solution but it is not. When we start using intermediate implications, combinatorial explosion is uncontrollable and T&E often slips in surreptitiously.