The New Sudoku Players' Forum

[denis_berthier]

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

5.......8.1.....4...2...6....87.65..725498361..65.14....9...8...6.....7.3.......2 # 95135 FNBYK C29/S4.hv
29 givens, SER = 7.3

[marek stefanik]

We have no given in b28, which gives birth to an interesting uniqueness technique.
We cannot have rope pattern in the middle stack, otherwise we get another solution by changing its direction.

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   +---------------------+---------------------+---------------------+
   | 5      79    b34–7  | 236–1  1467   2347  | 27     19     8     |
   | 6      1     c37    | 289–3  578    2579–3| 27     4     d35    |
   |a48     789    2     |f13     1457   457–3 | 6      19    e35    |
   +---------------------+---------------------+---------------------+
   | 1      4      8     | 7      3      6     | 5      2      9     |
   | 7      2      5     | 4      9      8     | 3      6      1     |
   | 9      3      6     | 5      2      1     | 4      8      7     |
   +---------------------+---------------------+---------------------+
   |h24     5      9     |g12     147    47–2  | 8      3      6     |
   | 28     6      1     | 238    58     235   | 9      7      4     |
   | 3      78     47    | 689    468    49    | 1      5      2     |
   +---------------------+---------------------+---------------------+

4r3c1 = (4–3)r1c3 = r2c3 – r2c9 = r3c9 – (3=1)r3c4 – (1=2)r7c4 – (2=4)r7c1 – Loop => –1r1c4, –2r7c6, –3r2c46, –3r3c6, –7r1c3

Code: Select all

   +------------------+------------------+------------------+
   | 5     79    34   | 36–2  1467  2347 | 27    19    8    |
   | 6     1     37   | 89–2  578   2579 | 27    4     35   |
   | 48    789   2    |b13    1457  457  | 6     19    35   |
   +------------------+------------------+------------------+
   | 1     4     8    | 7     3     6    | 5     2     9    |
   | 7     2     5    | 4     9     8    | 3     6     1    |
   | 9     3     6    | 5     2     1    | 4     8     7    |
   +------------------+------------------+------------------+
   | 24    5     9    |b12    147   47   | 8     3     6    |
   |a28    6     1    |b238   58    35–2 | 9     7     4    |
   | 3     78    47   | 689   468   49   | 1     5     2    |
   +------------------+------------------+------------------+

(2=8)r8c1 – (8=123)r378c4 => –2r8c6; 2b8\c4 => –2r12c4

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   +------------------+------------------+------------------+
   | 5     79   A34   | 36    1467 *27+34|*27    19    8    |
   | 6     1   hB37   |d8+#9 c578  *27+59|*27    4   iC3–5  |
   | 48    789   2    | 13    1457  457  | 6     19    35   |
   +------------------+------------------+------------------+
   | 1     4     8    | 7     3     6    | 5     2     9    |
   | 7     2     5    | 4     9     8    | 3     6     1    |
   | 9     3     6    | 5     2     1    | 4     8     7    |
   +------------------+------------------+------------------+
   |f24    5     9    |e#2+1  147   47   | 8     3     6    |
   | 28    6     1    |#23+8 b58   a35   | 9     7     4    |
   | 3     78   g47   |#689   468   49   | 1     5     2    |
   +------------------+------------------+------------------+

UR 27r12c67 using internals, 239r789c4 using mixed internals and externals:
3r1c6 – (3=5)r8c6 – (5=8)r8c5 – 8r2c5 = 8r2c4 – [9r2c4 | 8r8c4] = (1–2)r7c4 = (2–4)r7c1 = (4–7)r9c3 = (7–3)r2c3 = 3r2c9
4r1c6 – (4=3)r1c3 – 3r2c3 = 3r2c9
5r2c6
9r2c6 – (9=8)r2c4(d) – [9r2c4 | 8r8c4] = (1–2)r7c4 = (2–4)r7c1 = (4–7)r9c3 = (7–3)r2c3 = 3r2c9
=> –5r2c9, stte

Marek

PS: Does anyone know what the deadly rope pattern is called?

[eleven]

Do you mean, that with a "rope pattern" in a solution you could re-arrange the minirows in b28 (the upper and lower minirows change their column), so that the stack gets a second solution - independent of the other stacks ?

Never heard of this nice observation.

[marek stefanik]

Yes, exactly.

Edit: Minicolumns, not minirows. Other than that it's precisely what I meant.

[denis_berthier]

We have no given in b28, which gives birth to an interesting uniqueness technique.
We cannot have rope pattern in the middle stack, otherwise we get another solution by changing its direction.

I can't see in your resolution path where you're using this "rope pattern". Could you be more precise?

[marek stefanik]

In the last step, the part 8r2c4 – [9r2c4 | 8r8c4] = (1–2)r7c4.

Added: A simpler deduction for demonstration purposes:

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   +---------------------+---------------------+---------------------+
   | 5      79     347   | 1236   1467   2347  | 27     19     8     |
   | 6      1      37    |bc238+*9 a578  23579 | 27     4      35    |
   | 48     789    2     |e13     1457   3457  | 6      19     35    |
   +---------------------+---------------------+---------------------+
   | 1      4      8     | 7      3      6     | 5      2      9     |
   | 7      2      5     | 4      9      8     | 3      6      1     |
   | 9      3      6     | 5      2      1     | 4      8      7     |
   +---------------------+---------------------+---------------------+
   | 24     5      9     |d*2+1   147    247   | 8      3      6     |
   |f28     6      1     |cf*23+8 5–8    235   | 9      7      4     |
   | 3      78     47    |*689    468    49    | 1      5      2     |
   +---------------------+---------------------+---------------------+

8r2c5 = 8r2c4 – [9r2c4 | 8r8c4] = 1r7c4 – (1=3)r3c4 – (3=28)r8c14 => –8r8c5

[denis_berthier]

In the last step, the part 8r2c4 – [9r2c4 | 8r8c4] = (1–2)r7c4.

Added: A simpler deduction for demonstration purposes:

Code: Select all

   +---------------------+---------------------+---------------------+
   | 5      79     347   | 1236   1467   2347  | 27     19     8     |
   | 6      1      37    |bc238+*9 a578  23579 | 27     4      35    |
   | 48     789    2     |e13     1457   3457  | 6      19     35    |
   +---------------------+---------------------+---------------------+
   | 1      4      8     | 7      3      6     | 5      2      9     |
   | 7      2      5     | 4      9      8     | 3      6      1     |
   | 9      3      6     | 5      2      1     | 4      8      7     |
   +---------------------+---------------------+---------------------+
   | 24     5      9     |d*2+1   147    247   | 8      3      6     |
   |f28     6      1     |cf*23+8 5–8    235   | 9      7      4     |
   | 3      78     47    |*689    468    49    | 1      5      2     |
   +---------------------+---------------------+---------------------+

8r2c5 = 8r2c4 – [9r2c4 | 8r8c4] = 1r7c4 – (1=3)r3c4 – (3=28)r8c14 => –8r8c5

I still can't see any logic in the [9r2c4 | 8r8c4] = 1r7c4 part

[denis_berthier]

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Resolution state after Singles and whips[1]:
   +-------------------+-------------------+-------------------+
   ! 5     79    347   ! 1236  1467  2347  ! 27    19    8     !
   ! 6     1     37    ! 2389  578   23579 ! 27    4     35    !
   ! 48    789   2     ! 13    1457  3457  ! 6     19    35    !
   +-------------------+-------------------+-------------------+
   ! 1     4     8     ! 7     3     6     ! 5     2     9     !
   ! 7     2     5     ! 4     9     8     ! 3     6     1     !
   ! 9     3     6     ! 5     2     1     ! 4     8     7     !
   +-------------------+-------------------+-------------------+
   ! 24    5     9     ! 12    147   247   ! 8     3     6     !
   ! 28    6     1     ! 238   58    235   ! 9     7     4     !
   ! 3     78    47    ! 689   468   49    ! 1     5     2     !
   +-------------------+-------------------+-------------------+

The puzzle has an easy solution in Z5.

It also has 2-step solutions, but they require much harder steps. Here are the simplest two:

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whip[7]: r2n9{c4 c6} - r9n9{c6 c4} - c4n6{r9 r1} - b2n2{r1c4 r1c6} - r1n3{c6 c3} - c3n4{r1 r9} - r9c6{n4 .} ==> r2c4 ≠ 8
singles ==> r2c5 = 8, r8c5 = 5
whip[8]: r7c1{n2 n4} - c3n4{r9 r1} - c3n3{r1 r2} - r2c9{n3 n5} - r3n5{c9 c6} - c6n4{r3 r9} - r9n9{c6 c4} - r2c4{n9 .} ==> r7c4 ≠ 2
stte

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z-chain[8]: r9c6{n4 n9} - c4n9{r9 r2} - r2n8{c4 c5} - r8c5{n8 n5} - r3n5{c5 c9} - r2c9{n5 n3} - c3n3{r2 r1} - r1n4{c3 .} ==> r3c6 ≠ 4
whip[8]: r7c1{n2 n4} - c3n4{r9 r1} - c6n4{r1 r9} - r9n9{c6 c4} - r2n9{c4 c6} - c6n2{r2 r1} - r1n3{c6 c4} - c4n6{r1 .} ==> r7c4 ≠ 2
stte

[marek stefanik]

I still can't see any logic in the [9r2c4 | 8r8c4] = 1r7c4 part

If you were to delete 9r2c4, 8r8c4 and 1r7c4, you would end up with 239r789c4, forcing the rope pattern.
But if there were a solution with the rope pattern, you could just rearrange the minicolumns in b28 and get a different one.
Therefore at least one of them is true (assuming there is only one solution).

[denis_berthier]

I still can't see any logic in the [9r2c4 | 8r8c4] = 1r7c4 part

If you were to delete 9r2c4, 8r8c4 and 1r7c4, you would end up with 239r789c4, forcing the rope pattern.
But if there were a solution with the rope pattern, you could just rearrange the minicolumns in b28 and get a different one.
Therefore at least one of them is true (assuming there is only one solution).

delete n1r7c4 or assume it?

And what's the rope pattern here?

[eleven]

Code: Select all

+-------------------+-------------------+-------------------+
| 5     79    34    | 36    1467  2347  | 27    19    8     |
| 6     1     37    | 8+9   57+8  2579  | 27    4     35    |
| 48    789   2     | 13    1457  457   | 6     19    35    |
+-------------------+-------------------+-------------------+
| 1     4     8     | 7     3     6     | 5     2     9     |
| 7     2     5     | 4     9     8     | 3     6     1     |
| 9     3     6     | 5     2     1     | 4     8     7     |
+-------------------+-------------------+-------------------+
| 24    5     9     | 12    147   47    | 8     3     6     |
| 28    6     1     | 23+8  58    35    | 9     7     4     |
| 3     78    47    | 9+68  468   49    | 1     5     2     |
+-------------------+-------------------+-------------------+

In the minicolumn r456c5 there is 392. If you would have 239 in the minicolumn r789c4 too, it is forced to be also in r123c6, and in r789c6 there is left 457, as well as in (r456c4 and) r123c5, and r789c5 must be 168, as well as r123c4.
That's the (invalid) rope pattern (change the columns of the minicolums of boxes 28 in a solution to get another solution).
With 8r2c4 you get 9r9c4, and 23 in r8c4. So if r7c4 is not 1, you get the rope pattern with 239 in r789c4.

[Added:]So note, that none of the mini-columns in b28 can have the same digits as in the minicolums of box 5.

[marek stefanik]

delete n1r7c4 or assume it?

I'm not completely sure what you're referring to.
If you deleted it together with the other guardians, you'd get a contradiction, so if you delete the other guardians (which 8r2c4 does), you can assume it.

[denis_berthier]

eleven,
thanks for the explanation
that was totally absent from marek's notation.

This uniqueness pattern must be quite rare.

[shye]

probably rare yeah, requires two empty boxes in a band/stack and that doesnt guarantee itll always be useful, really cool find!