The New Sudoku Players' Forum

[marek stefanik]

Code: Select all

   +---------+---------+---------+
   | .  3  . | .  .  6 | 5  .  8 |
   | .  4  2 | .  .  7 | .  6  . |
   | 5  .  . | .  .  . | .  .  2 |
   +---------+---------+---------+
   | 6  .  . | 7  .  . | .  .  . |
   | .  7  . | .  2  . | .  1  . |
   | .  .  8 | .  .  9 | 3  .  . |
   +---------+---------+---------+
   | .  .  9 | .  .  3 | 7  .  . |
   | .  .  . | .  1  . | .  .  . |
   | 2  .  . | 8  .  . | .  .  . |
   +---------+---------+---------+
.3...65.8.42..7.6.5.......26..7......7..2..1...8..93....9..37......1....2..8.....

A 24-clue minimal.
Merry Christmas!

[denis_berthier]

.

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 179    3      17     ! 2      49     6      ! 5      479    8      !
   ! 89     4      2      ! 135    3589   7      ! 19     6      139    !
   ! 5      689    67     ! 134    3489   148    ! 149    3479   2      !
   +----------------------+----------------------+----------------------+
   ! 6      159    1345   ! 7      3458   1458   ! 2      4589   459    !
   ! 349    7      345    ! 345    2      458    ! 4689   1      4569   !
   ! 14     2      8      ! 1456   456    9      ! 3      45     7      !
   +----------------------+----------------------+----------------------+
   ! 148    158    9      ! 456    456    3      ! 7      2      145    !
   ! 347    56     34567  ! 9      1      2      ! 468    3458   3456   !
   ! 2      156    13456  ! 8      7      45     ! 1469   3459   134569 !
   +----------------------+----------------------+----------------------+
159 candidates.

Simplest-first solution in W7: Show

z-chain[4]: r8n7{c3 c1} - c1n3{r8 r5} - b4n9{r5c1 r4c2} - c2n5{r4 .} ==> r8c3≠5
finned-swordfish-in-columns: n5{c3 c6 c9}{r4 r5 r9} ==> r9c8≠5
whip[4]: r6c8{n5 n4} - r6c1{n4 n1} - r4c2{n1 n9} - r4c9{n9 .} ==> r4c8≠5
biv-chain[5]: r4n3{c5 c3} - c1n3{r5 r8} - b7n7{r8c1 r8c3} - r3n7{c3 c8} - b3n3{r3c8 r2c9} ==> r2c5≠3
z-chain[5]: r1n1{c3 c1} - c1n7{r1 r8} - c1n3{r8 r5} - b4n9{r5c1 r4c2} - c2n1{r4 .} ==> r9c3≠1
whip[5]: r6c8{n5 n4} - r1n4{c8 c5} - r6c5{n4 n6} - r7c5{n6 n5} - r2n5{c5 .} ==> r6c4≠5
whip[5]: r1n4{c8 c5} - b8n4{r7c5 r7c4} - c4n6{r7 r6} - r6n1{c4 c1} - r6n4{c1 .} ==> r9c8≠4
t-whip[6]: b3n3{r2c9 r3c8} - r3n7{c8 c3} - r3n6{c3 c2} - r8c2{n6 n5} - r9c2{n5 n1} - r7n1{c2 .} ==> r2c9≠1
whip[1]: c9n1{r9 .} ==> r9c7≠1
whip[6]: c8n7{r3 r1} - b3n4{r1c8 r3c7} - r3c4{n4 n1} - r3c6{n1 n8} - c5n8{r3 r4} - c5n3{r4 .} ==> r3c8≠3
hidden-single-in-a-block ==> r2c9=3
z-chain[7]: c5n3{r4 r3} - c5n8{r3 r2} - b1n8{r2c1 r3c2} - c2n9{r3 r4} - r4c9{n9 n5} - r6c8{n5 n4} - r1n4{c8 .} ==> r4c5≠4
whip[6]: r6c8{n5 n4} - r6c1{n4 n1} - r4n1{c3 c6} - r4n4{c6 c3} - c3n5{r4 r9} - c6n5{r9 .} ==> r5c9≠5
whip[7]: r6c1{n4 n1} - r4n1{c3 c6} - r4n4{c6 c3} - r4n3{c3 c5} - r3n3{c5 c4} - r3n1{c4 c7} - b3n4{r3c7 .} ==> r6c8≠4
naked-single ==> r6c8=5
z-chain[5]: c2n6{r9 r3} - c2n9{r3 r4} - r4c9{n9 n4} - r8c9{n4 n5} - r8c2{n5 .} ==> r8c3≠6
whip[5]: r4c9{n4 n9} - b4n9{r4c2 r5c1} - c1n3{r5 r8} - r8n7{c1 c3} - r8n4{c3 .} ==> r7c9≠4
biv-chain[3]: r9n1{c2 c9} - r7c9{n1 n5} - r8n5{c9 c2} ==> r9c2≠5
t-whip[4]: r9c6{n5 n4} - r7n4{c5 c1} - r6c1{n4 n1} - b5n1{r6c4 .} ==> r4c6≠5
z-chain[5]: r6n4{c5 c1} - r7n4{c1 c5} - r9c6{n4 n5} - b5n5{r5c6 r4c5} - b5n3{r4c5 .} ==> r5c4≠4
biv-chain[4]: r6n1{c1 c4} - r2c4{n1 n5} - r5c4{n5 n3} - r4n3{c5 c3} ==> r4c3≠1
hidden-single-in-a-column ==> r1c3=1
z-chain[4]: c3n7{r8 r3} - c3n6{r3 r9} - b7n3{r9c3 r8c1} - r8n7{c1 .} ==> r8c3≠4
z-chain[4]: r8n4{c9 c1} - c1n3{r8 r5} - b4n9{r5c1 r4c2} - r4c9{n9 .} ==> r9c9≠4
t-whip[4]: c9n4{r5 r8} - r8n5{c9 c2} - r8n6{c2 c7} - c7n8{r8 .} ==> r5c7≠4
t-whip[4]: c6n5{r9 r5} - r5c4{n5 n3} - b4n3{r5c1 r4c3} - c3n5{r4 .} ==> r9c9≠5
biv-chain[4]: r9c2{n6 n1} - b9n1{r9c9 r7c9} - c9n5{r7 r8} - r8c2{n5 n6} ==> r3c2≠6, r9c3≠6
singles ==> r3c3=6, r1c1=7, r3c8=7, r8c3=7
biv-chain[4]: c1n9{r5 r2} - r2n8{c1 c5} - r2n5{c5 c4} - r5c4{n5 n3} ==> r5c1≠3
singles ==> r8c1=3, r9c8=3
whip[1]: r8n4{c9 .} ==> r9c7≠4
finned-x-wing-in-columns: n9{c2 c8}{r4 r3} ==> r3c7≠9
biv-chain[4]: b4n3{r4c3 r5c3} - r5c4{n3 n5} - c6n5{r5 r9} - r9n4{c6 c3} ==> r4c3≠4
z-chain[4]: c4n6{r6 r7} - c4n4{r7 r3} - r3c7{n4 n1} - c6n1{r3 .} ==> r6c4≠1
singles ==> r4c6=1, r6c1=1
whip[1]: r6n4{c5 .} ==> r5c6≠4
whip[1]: b4n4{r5c3 .} ==> r5c9≠4
biv-chain[3]: c9n4{r4 r8} - r8n5{c9 c2} - r4c2{n5 n9} ==> r4c9≠9
naked-single ==> r4c9=4
naked-pairs-in-a-row: r8{c2 c9}{n5 n6} ==> r8c7≠6
swordfish-in-columns: n9{c1 c7 c9}{r5 r2 r9} ==> r2c5≠9
biv-chain[3]: r2c5{n5 n8} - r3c6{n8 n4} - r9c6{n4 n5} ==> r7c5≠5
naked-pairs-in-a-column: c5{r6 r7}{n4 n6} ==> r3c5≠4, r1c5≠4
stte

2-step solution, using the nukes: Show

FORCING[3]-T&E(W1) applied to trivalue candidates n1r7c1, n4r7c1 and n8r7c1 :
===> 3 values decided in the three cases: n6r3c3 n7r3c8 n3r2c9
===> 22 candidates eliminated in the three cases: n9r1c1 n7r1c8 n3r2c4 n3r2c5 n9r2c5 n1r2c9 n9r2c9 n6r3c2 n7r3c3 n8r3c5 n3r3c8 n4r3c8 n9r3c8 n9r4c9 n5r5c4 n5r6c4 n5r8c3 n6r8c3 n3r8c9 n1r9c3 n6r9c3 n3r9c9

whip[1]: c9n1{r9 .} ==> r9c7≠1

FORCING[3]-T&E(W1) applied to trivalue candidates n3r3c5, n4r3c5 and n9r3c5 :
===> 10 values decided in the three cases: n5r6c8 n4r4c9 n3r9c8 n5r8c9 n6r8c2 n1r7c9 n1r9c2 n8r7c2 n5r9c3 n4r9c6
===> 49 candidates eliminated in the three cases: n8r3c2 n1r3c4 n4r3c6 n1r4c2 n4r4c3 n5r4c3 n4r4c5 n5r4c5 n4r4c6 n4r4c8 n5r4c8 n5r4c9 n4r5c1 n5r5c3 n4r5c6 n4r5c7 n4r5c9 n5r5c9 n4r6c4 n5r6c5 n4r6c8 n1r7c1 n8r7c1 n1r7c2 n5r7c2 n4r7c4 n4r7c5 n4r7c9 n5r7c9 n4r8c1 n5r8c2 n4r8c3 n6r8c7 n3r8c8 n5r8c8 n4r8c9 n6r8c9 n5r9c2 n6r9c2 n3r9c3 n4r9c3 n5r9c6 n4r9c7 n4r9c8 n5r9c8 n9r9c8 n1r9c9 n4r9c9 n5r9c9

stte

[DEFISE]

As I'm not an expert in tricks, here is a relatively short solution in W10:

Hidden Text: Show

Singles: 2r1c4, 2r7c8, 2r6c2, 2r4c7, 2r8c6, 7r9c5, 7r6c9, 9r8c4
Box/Line: 1r1b1 => -1r2c1 -1r3c2 -1r3c3
Box/Line: 6r6b5 => -6r5c4
Box/Line: 8r7b7 => -8r8c1 -8r8c2
Box/Line: 6b8r7 => -6r7c2 -6r7c9

whip[10]: r1c3{n7 n1}- r1n7{c3 c8}- r1n4{c8 c5}- r1n9{c5 c1}- c2n9{r3 r4}- r4n1{c2 c6}- r3c6{n1 n8}- r5n8{c6 c7}- r5n6{c7 c9}- r5n9{c9 .} => -7r3c3

Singles: 6r3c3, 7r3c8, 3r2c9
Box/Line: 1c9b9 => -1r9c7
Naked pairs: 49r1c58 => -9r1c1

whip[10]: b3n4{r1c8 r3c7}- c7n1{r3 r2}- r2c4{n1 n5}- r6n5{c4 c5}- r6n6{c5 c4}- b5n1{r6c4 r4c6}- r3c6{n1 n8}- c5n8{r2 r4}- r4n3{c5 c3}- r4n4{c3 .} => -4r6c8

Single: 5r6c8

whip[10]: b4n9{r5c1 r4c2}- r3c2{n9 n8}- r2n8{c1 c5}- r2n5{c5 c4}- r5c4{n5 n4}- b5n3{r5c4 r4c5}- c5n5{r4 r7}- r7c2{n5 n1}- r7c9{n1 n4}- r4c9{n4 .} => -3r5c1

Singles: 3r8c1, 7r8c3, 1r1c3, 7r1c1, 3r9c8
Box/Line: 4r8b9 => -4r7c9 -4r9c7 -4r9c9
Naked pairs: 45r9c36 => -5r9c2 -5r9c9

whip[9]: r1c5{n9 n4}- r6c5{n4 n6}- r7c5{n6 n5}- c9n5{r7 r8}- c2n5{r8 r4}- c6n5{r4 r5}- r5n8{c6 c7}- r4c8{n8 n4}- c9n4{r4 .} => -9r1c8

STTE

[marek stefanik]

Thank you for your solutions.

I think they show that the puzzle is quite persistent when attacked with chains.

To give a hint about what the trick is, it is the puzzle's minimality that is to be exploited.

Marek

[eleven]

Ah ok ...
Since it is minimal, there must be another solution, when i remove 2r2c3.

Code: Select all

+-------------------------+-------------------------+-------------------------+
| 179     3       127     | 1249    49      6       | 5       479     8       |
| 189     4       12      | 12359   3589    7       | 19      6       139     |
| 5       1689    167     | 1349    3489    148     | 149     3479    2       |
+-------------------------+-------------------------+-------------------------+
| 6       1259    12345   | 7       3458    1458    | 2489    24589   459     |
| 349     7       345     | 3456    2       458     | 4689    1       4569    |
| 14      125     8       | 1456    456     9       | 3       2457    4567    |
+-------------------------+-------------------------+-------------------------+
| 148     1568    9       | 2456    456     3       | 7       2458    1456    |
| 3478    568     34567   | 24569   1       245     | 24689   234589  34569   |
| 2       156     134567  | 8       45679   45      | 1469    3459    134569  |
+-------------------------+-------------------------+-------------------------+

So i can set r2c3 to 1 and try to solve it. This is a singles puzzle with the solution

Code: Select all

+-------+-------+-------+
| 7 3 2*|*1 9 6 | 5 4 8 |
| 8 4 1*|*2 5 7 | 9 6 3 |
| 5 9 6 | 4 3 8 | 1 7 2 |
+-------+-------+-------+
| 6 5 3 | 7 8 1 | 2 9 4 |
| 9 7 4 | 3 2 5 | 8 1 6 |
| 1 2 8 | 6 4 9 | 3 5 7 |
+-------+-------+-------+
| 4 8 9 | 5 6 3 | 7 2 1 |
| 3 6 7 | 9 1 2 | 4 8 5 |
| 2 1 5 | 8 7 4 | 6 3 9 |
+-------+-------+-------+

To get the solution of the original puzzle, just switch 1 and 2 in r12c34.

[yzfwsf]

Thank you for your solutions.

I think they show that the puzzle is quite persistent when attacked with chains.

To give a hint about what the trick is, it is the puzzle's minimality that is to be exploited.

Marek

Your hint, or UA4, is more accurate

[marek stefanik]

Yes, that's it, you can start with the UR like this:

Code: Select all

+-------------------------+-------------------------+-------------------------+
| 179     3      #127     |#1249    49      6       | 5       479     8       |
| 189     4      #12      |#12359   3589    7       | 19      6       139     |
| 5       1689    167     | 1349    3489    148     | 149     3479    2       |
+-------------------------+-------------------------+-------------------------+
| 6       1259    12345   | 7       3458    1458    | 2489    24589   459     |
| 349     7       345     | 3456    2       458     | 4689    1       4569    |
| 14      125     8       | 1456    456     9       | 3       2457    4567    |
+-------------------------+-------------------------+-------------------------+
| 148     1568    9       | 2456    456     3       | 7       2458    1456    |
| 3478    568     34567   | 24569   1       245     | 24689   234589  34569   |
| 2       156     134567  | 8       45679   45      | 1469    3459    134569  |
+-------------------------+-------------------------+-------------------------+

There has to be a solution without 2r2c3, but every such solution (NS r2c3, HSs 1r1, 2b12) contains the UR, meaning there is a solution with the UR.
The two solutions of the UR are equivalent, so there is a solution with the UR and 2r2c3, which obviously is a solution to the original puzzle.
So setting 12r12c34, you are still guaranteed a solution. stte

Marek

[denis_berthier]

Since it is minimal, there must be another solution, when i remove 2r2c3.[...]
To get the solution of the original puzzle, just switch 1 and 2 in r12c34.

Using minimality is a smart idea.
However, this is not solving the puzzle; this is solving the puzzle with an oracle (the knowledge that it is minimal) - which can completely change its complexity. But a general well-formed puzzle is not guaranteed to be minimal.