## Some nice toughies...

Advanced methods and approaches for solving Sudoku puzzles
ronk wrote:Would someone please tell me how to get past this point?

This is my solution. Points 3 and 6 are rather complicated.

After the swordfish in 8 r479c357 => r479c28<>8, r9c9<>8:
[edited typo]
1.r4c3=8 => r6c8=8 => r2c9=8 => r2c2=2 => no 2 for col3 => r6c2=8
2.
r6c8=9 => r6c1=5
r6c8=9 => r5c7=7 => r5c9=3 => r5c3=5
=> r5c7=9
Brings us here with an type 4 UR 68 in r79c35 (r79c3<>6)
Code: Select all
`3567   9       23456   |   8      17     24     |  2567     1357    2367       57     257     1       |   6      3      9      |  4        578     278        8      23457   23456   |   5      17     24     |  267      137     9          -------------------------------------------------------------------------   4      123     23      |   17     9      6      |  8        37      5          135    6       35      |   17     4      8      |  9        2       37         9      8       7       |   2      5      3      |  1        6       4          -------------------------------------------------------------------------   2      3457    3458    |   39     68     57     |  567      49      1          567    57      9       |   4      2      1      |  3        578     678        13567  13457   3458    |   39     68     57     |  2567     49      267`
----
3.
r4c2=1 => r9c2<>1
r4c2=1 => r4c4=7 => r3c8=3 => r1c9=3 => r13c7=6 => r79c6<>6 => (AUR 57 in r79c67)r9c7=2 => r2c9=2 => r28c2=57
=> r79c2=34 => deadly pattern 34/39/49 in r79c248
=> r5c1=1
4. ER in 5: r28c8,box 1 => r8c1<>5 => r8c1=6
5. xy-chain: 7-r8c2-5-r8c8-8-r2c8-5-r1c2-7 => r89c1<>7,r23c2<>7
6.
r9c1=5 => r1c1=3 => r1c8=1 => r1c5=7
r9c1=5 => r2c1=7,r2c2=5 => r2c8=8 => r2c9=2 => r1c7=2 (=> r9c9<>2)
r9c1=5 => r9c6=7 => r9c9=6
=> r1c9 is empty => r9c1=3
Last edited by ravel on Sat May 20, 2006 5:56 am, edited 1 time in total.
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:
ronk wrote:Would someone please tell me how to get past this point?

This is my solution. Points 3 and 6 are rather complicated.

After the swordfish in 8 r479c357 => r479c28<>8, r9c9<>8:
1.r4c3=8 => r6c8=8 => r2c9=8 => r2c2=2 => no 2 for col3 => r5c2=8

something's amiss
[52]=6 is a clue from ronk's pencilmarks
gsf
2014 Supporter

Posts: 7306
Joined: 21 September 2005
Location: NJ USA

gsf wrote:[52]=6 is a clue from ronk's pencilmarks

Thanks, should be r6c2=8, i correct it above.
ravel

Posts: 998
Joined: 21 February 2006

This one should not be too hard but is interesting:
#7:
Code: Select all
`8...........3..92.....21..5..8.1...7.6.5..1...9..3.2.65..8.......4..3.6...1..6..9`

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

Posts: 26
Joined: 14 May 2006

This one is #276 from Into Sudoku - posted by Scott. Into Sudoku needs many forcing chains and has to guess 4x:
#8
Code: Select all
`...86..1.7....1..5......8..2...1.7.81......4..5.7.4..9.73.....6.2...5...6....9..7`

Code: Select all
`. . .|8 6 .|. 1 .7 . .|. . 1|. . 5. . .|. . .|8 . .-----+-----+-----2 . .|. 1 .|7 . 81 . .|. . .|. 4 .. 5 .|7 . 4|. . 9-----+-----+-----. 7 3|. . .|. . 6. 2 .|. . 5|. . .6 . .|. . 9|. . 7`
Karlson

Posts: 26
Joined: 14 May 2006

I also needed 4 steps, will add it to my list
ravel

Posts: 998
Joined: 21 February 2006

Here is a possible solution for Puzzle #4:

Code: Select all
`*--------------------------------------------------------------------* | 2      3      5689   | 5679   1456   46789  | 1459   1456   469    | | 5689   568    1      | 3569   2      34689  | 459    3456   7      | | 7      4      569    | 3569   1356   369    | 8      12356  2369   | |----------------------+----------------------+----------------------| | 4      568    35678  | 236    9      1      | 25     2357   23     | | 369    2      3679   | 4      36     5      | 19     137    8      | | 359    1      359    | 8      7      23     | 6      2345   2349   | |----------------------+----------------------+----------------------| | 356    7      23456  | 1      8      2346   | 24     9      246    | | 1      68     2468   | 79     46     79     | 3      2468   5      | | 368    9      23468  | 2356   3456   2346   | 7      2468   1      | *--------------------------------------------------------------------*`

1. [r6c6]-3-[r7c6|r8c5|r9c6]-2-[r6c6], => r2c6/r3c6<>3.

2. [r6c9]-2/3-[r3c6|r34c9]-6,9-[r3c3]-5-[r2c2]=5=[r4c2](-5-[r4c7]-2-
[r6c9])-5-[r6c13]-3-[r6c9], =>r6c9<>2,3.

3. [r4c4]-3-[r2c4]=3=[r2c8]-3-[r3c9]=3=[r4c9]-3-[r4c4], => r4c4<>3.

4. [r7c6]-2-[r7c7]=2=[r4c7]-2-[r4c4]=2=[r6c6]-2-[r7c6], => r7c6<>2.

5. [r5c8]-3-[r5c5]-6-[r4c4]-2-[r4c9]-3-[r5c8], => r5c8<>3.

6. [r6c6](-3-[r5c5]-6-[r8c5]-4-[r7c6])-3-[r6c13](-9-[r6c9]-4-[r7c9])-5-
[r4c2]=5=[r2c2]-5-[r3c36]-6,9-[r34c9]-2-[r7c9]-6-[r7c6]-3-[r6c6],
=> r6c6<>3.

7. [r1c5]-1-[r1c78]=1=[r3c8]-1-[r5c8]-7-[r5c3]=7=[r4c3]=8=[r4c2]=
=5=[r2c2]-5-[r3c3]=5|1=[r3c5]-1-[r1c5], => r1c5<>1.

8. [r1c9]-6-[r1c5]=6=[r8c5](-6-[r8c8])-6-[r8c2]-8-[r7c7|r8c8]-2,4-[r7c9]-
-6-[r1c9], => r1c9<>6.

9. [r4c8]=7=[r5c8]=1=[r5c7]=9=[r6c9]=4=[r6c8]-4-[r89c8]=4=[r7c7]=
=2=[r4c7](-2-[r4c8])-2-[r4c9]-3-[r4c8], => r4c8<>2,3.

10. [r1c46]-9-[r1c9]-4-[r1c5]-6-[r3c6]-9-[r1c46], => r1c4/r1c6<>9.

11. [r2c8](-3-[r3c9]=3=[r4c9]=2=[r4c7]-2-[r7c7]-4-[r2c7])-3-
[r2c47](-5-[r2c2]=5=[r4c2]-5-[r6c1])-5,9-[r2c12]=5,9=[r13c3]-5,9-[r6c3]
-3-[r6c1]-9-[r6c9]=9=[r1c9](-9-[r2c7])-9-[r1c3]-5-[r1c4]-7-[r8c4]
-9-[r2c4]-5-[r2c7], => r2c8<>3.

12. [r2c8]-5-[r2c2]=5=[r4c2]-5-[r6c13]=5=[r6c8]-5-[r2c8], => r2c8<>5.

13. [r2c1]-5-[r7c1]=5=[r7c3]=2=[r7c7]-2-[r4c7]-5-[r4c2]=5=[r2c2]-
-5-[r2c1], => r2c1<>5.

14. [r1c7]=1=[r5c7]=9=[r6c9]=4=[r1c9]-4-[r1c7], => r1c7<>4.

15. [r7c6]=3=[r9c6]=6=[r8c5]-6-[r8c2]=6=[r2c2]=5=[r2c7]=4=[r7c7]-
-4-[r7c6], => r7c6<>4.

16. [r9c1]=3=[r6c1]-3-[r6c8]=3=[r4c9]=2=[r4c7]-2-[r7c7]-4-[r9c8]-
-8-[r9c1], => r9c1<>8.

17. [r5c1]=6=[r5c3]=7=[r5c8]=1=[r5c7]=9=[r6c9]=4=[r1c9]-4-[r1c5]=
=4=[r8c5]=6=[r9c6]-6-[r9c1]=6=[r5c1], => r5c1=6 which solve the puzzle.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

A possible solution for Puzzle #2.

This puzzle allows some interesting applications of the TILA (Two Incompatible Loops Argument) and of the corresponding Almost TILA (ATILA). This post explains the TILA.

Code: Select all
` *----------------------------------------------------------* | 2349  23469   8     | 79    147  1349  | 479  5    14679 | | 7     49      5     | 6     2    149   | 8    3    149   | | 1     3469    469   | 5789  478  3459  | 2    469  4679  | |---------------------+------------------+-----------------| | 349   34789   49    | 1     5    2     | 6    489  479   | | 259   12589   129   | 4     6    7     | 3    189  25    | | 245   124567  1246  | 3     9    8     | 47   14   25    | |---------------------+------------------+-----------------| | 2459  12459   3     | 2589  148  14569 | 49   7    4689  | | 6     459     7     | 589   3    459   | 1    2    489   | | 8     1249    1249  | 279   147  1469  | 5    469  3     | *----------------------------------------------------------*`

1. Type-3 Unique Rectangle in cells [r56c19|r4c3].

2. [r4c8]=8=[r5c8]=9=[r5c1]-9-[r4c3]-4-[r4c8], => r4c8<>4.

3. [r56c1]-2-[r1c1]=2|6=[r3c3]-6-[r56c3]-2-[r56c1], => r5c1/r6c1<>2.

4. [r9c5](-1-[r7c5])(-1-[r9c23]=1|5=[r8c2]-5-[r8c4])-1-[r1c457]-4,9-
[r1c1]-2-[r7c17]-4-[r7c5]-8-[r8c4]-9-[r1c4]-7-[r9c4]=7=[r9c5],
=> r9c5<>1.

5. =5=[r7c6](-5-[r8c4])=6=[r9c6](-6-[r9c8]=6=[r3c8]-6-[r1c9|r3c3])=1=
[r7c5]-1-[r1c457]-4,7,9-[r1c9]-1-[r2c9]=(ATILA: r2c2|r3c3|r4c3|r4c9|
|r2c9)=1|7=[r4c9]-7-[r6c7]=7=[r1c7]-7-[r1c4]-9-[r8c4]-8-[r8c9]=8=
=[r7c9]=6=[r7c6], => r7c6<>5.

6. [r3c2]=3=[r3c6]=5=[r8c6]-5-[r28c2]-4,9-[r3c2], => r3c2<>4,9.

7. [r1c6]=3=[r1c2](=2=[r1c1]-2-[r7c1])-3-[r3c2](-6-[r3c9])-6-[r3c38]-
-4,9-[r3c9]-7-[r4c9]=7=[r6c7]-7-[r1c7]-{TILA: r3c3|r4c3|r6c1|r7c1|r7c7|
|r1c7|r3c8}, => r1c2<>3.

8. =7=[r1c9](-7-[r1c7]=7=[r6c7])=6=[r1c2](=2=[r1c1]-2-[r7c1])-6-
-[r3c3]=(ATILA: r3c3|r4c3|r6c1|r7c1|r7c7|r1c7|r3c8)=6=[r3c8]-6-[r3c9]-
-(ATILA: r3c3|r4c3|r6c1|r7c1|r7c7|r1c7|r3c9)-7-[r1c9], => r1c9<>7.

9. [r9c6]-1-[r9c23]=1=[r7c2]=5=[r7c4]=2=[r9c4]-2-[r9c23]=2=[r7c1]-
-2-[r1c1]=2=[r1c2]=6=[r3c3](-6-[r3c8]=6=[r9c8])-6-[r6c3]=6=[r6c2]=
=7=[r6c7]-7-[r1c7]-{TILA:r1c1|r1c7|r3c8|r6c8|r4c9|r4c3|r9c3|r9c2|r2c2}
=> r9c6<>1.

10. [r1c9]=1=[r1c5]-1-[r7c5]=1=[r7c6]=6=[r7c9]-6-[r1c9], => r1c9<>6.

11. [r6c7](-7-[r1c7])-7-[r4c9]=(ATILA: r2c9|r2c2|r3c3|r4c3|r4c9)=7|1=
=[r2c9]-1-[r1c9]-{TILA: r1c7|r1c9|r4c9|r4c3|r6c1|r7c1|r7c7},
=> r6c7<>7 and that solves the puzzle.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

Very sophisticated again.

Is it correct, that the ATILA in step 5 is a 49 remote pair and the TILA in step 7 a deadly bivalue/bilocation loop for number 4 (r6c1 containing 45, the rest 49) ?
If so, is it possible, that you denote in some way, what we can find in the (A)TILA ?
ravel

Posts: 998
Joined: 21 February 2006

Thanks.

Ravel wrote:Is it correct, that the ATILA in step 5 is a 49 remote pair

You can see it in that way, but it is easier to see it (for me) as an Almost Deadly Turbot Fish.

Ravel wrote:and the TILA in step 7 a deadly bivalue/bilocation loop for number 4 (r6c1 containing 45, the rest 49) ?

Yes, correct. For every cell C of that deadly pattern we can write two loops, one that forces C = 4 and another that implies C<>4: a contradiction situation.

Ravel wrote:If so, is it possible, that you denote in some way, what we can find in the (A)TILA ?

Yes. In this case, as every TILA is due to incompatible X-cycles on "4", it could be written "{TILA(4): ...}". In other cases where the TIL's are more complicated nice loops, I write "{TILA(x): (1); (2)}", where (1) and (2) are the two incompatible loops written in the known notation. How does that sound to you (and the others users)?
Carcul

Posts: 724
Joined: 04 November 2005

Puzzle #2: Scanning cells in descending order and only using exposed Naked/Hidden Singles in chains.

Code: Select all
`r4c5    =  5     Naked  Singler4c6    =  2     Naked  Singler5c5    =  6     Naked  Singler9c9    =  3     Hidden Singler5c7    =  3     Hidden Single    b3  -  1     Locked Candidate (1)    b9  -  8     Locked Candidate (1)  c9    -  25    Hidden Pairr9c8    =  6     [r9c8]=4 => [r7c6]=EMPTY ; [r9c8]=9 => [r7c1]=EMPTYr7c6    =  6     Hidden Singler7c7    =  9     [r7c7]=4 => [r5c8]=EMPTY  c9    -  48    Naked  Pairr1c7    ~  9     XY-Wingr4c9    ~  4     XY-Wingr6      -  25    Naked  Pairr6c8    =  1     [r6c8]=4 => [r8c2]=EMPTYr6c3    =  6     Naked/Hidden Singles complete puzzle`

Since I don't currently suport Uniqueness Rectangles, one of my steps above might be replaced by it -- as per Carcul.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Carcul wrote:Yes. In this case, as every TILA is due to incompatible X-cycles on "4", it could be written "{TILA(4): ...}". In other cases where the TIL's are more complicated nice loops, I write "{TILA(x): (1); (2)}", where (1) and (2) are the two incompatible loops written in the known notation. How does that sound to you (and the others users)?

Yes, good, it would be easier to follow then (sometimes i am not sure, if i still have the same candidates after some steps).
ravel

Posts: 998
Joined: 21 February 2006

A possible solution for Puzzle #6:

Code: Select all
` *---------------------------------------------------------------* | 35679  234579  23456 | 8    157   2457 | 2567    13567   2367 | | 57     257     1     | 6    3     9    | 4       578     278  | | 8      23457   23456 | 257  157   2457 | 2567    13567   9    | |----------------------+-----------------+----------------------| | 4      1238    238   | 127  9     267  | 678     3678    5    | | 1359   6       35    | 157  4     8    | 79      2       37   | | 59     2589    7     | 25   56    3    | 1       689     4    | |----------------------+-----------------+----------------------| | 2      34578   34568 | 39   5678  567  | 56789   456789  1    | | 567    578     9     | 4    2     1    | 3       5678    678  | | 13567  134578  34568 | 39   5678  567  | 256789  456789  2678 | *---------------------------------------------------------------*`

1. [r2c9]=8=[r2c8](-8-[r8c8])-8-[r6c8]=8=[r6c2]-8-[r8c2]=8=[r8c9]-8-
-[r2c9], => r479c28/r9c9<>8.

2. [r6c8]-8-[r4c7]=8=[r4c3]=2=[r4c2]-2-[r2c2]=2=[r2c9]=8=[r2c8]-8-
-[r6c8], => r6c8<>8.

3. [r13c5]-5-[r6c5]-6-[r4c6]=6=[r4c8]=3=[r5c9]-3-[r5c3]-5-[r5c4]=5=
=[r3c4]-5-[r13c5], => r1c5/r3c5<>5.

4. [r9c2]=1=[r9c1]-1-[r5c1]=1=[r5c4]=7=[r5c9]=3=[r1c9]-3-[r13c8]=
=(AUR: r13c58)=3|5=[r1c8]-5-[r28c8]=5=[X-Wing: r28c12]-5-[r9c2|r5c1]
=> r9c2/r5c1<>5.

5. [r5c1]=1=[r9c1]-1-[r9c2]=(AUPattern: r79c248/r8c2)=1|5,7=[r789c2]
(-5,7-[r2c2]-2-[r2c9])-5,7-[r8c1]-6-[r8c9]=(AUR: r28c89)=6|5=[r28c8]-
-5-[r1c8]=(AUR: r13c58)=5|3=[r13c8]-3-[r1c9]=3=[r5c9]-3-[r5c1],
=> r5c1<>3.

6. [r1c1]=5=[r1c7]-5-[r2c8]=5=[r8c8](-5-[r8c1]-5-[r1c1])=8=[r8c9]=6=
[r8c1]-6-[r1c1], => r1c1<>6; r8c1<>5.

7. [r23c2]-7-[r8c2]-5-[r8c8]-8-[r8c9]=8=[r2c9]=2=[r2c2]=7=[r2c1]-7-
-[r3c2], => r2c2/r3c2<>7.

8. [r3c7]=7=[r3c5]=1=[r3c8]=3=[X-Wing: r34c23](-3-[r9c3])-3-[r7c23]
=3=[r7c4]-3-[r9c4]-9-[r9c8]-4-[r9c3]-8-[r7c3]=8=[r7c5]=6=[r7c7]-6-
-[r3c7], => r3c7<>6.

9. [r9c9]-6-[r7c7]=6=[r7c5]=8=[r7c3]-8-[r9c348]-3-[r9c1]=3=[r1c1]=5=
=[r1c7](-5-[r7c7])=6=[r1c9]-6-[r9c9], => r9c9<>6; r7c7<>5.

10. [r9c9]=2=[r2c9]-2-[r2c2]-5-[r12c1]=5=[r9c1]-5-[r9c6]-7-[r9c9],
=> r9c9<>7 and that solves the puzzle.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

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