Challenge: New set of 11 'Unsolvables'

Advanced methods and approaches for solving Sudoku puzzles

Postby ravel » Thu Sep 21, 2006 10:45 am

StrmCkr wrote:they where published on a sub link... i dont see it either... werid..
Please can someone post them?
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Postby ronk » Thu Sep 21, 2006 10:57 am

Michael Mepham "Unsolveables" (sic) #23 thru #33:
Code: Select all
#23
...8.21...6..4..5.1......72..59.67......8......13.48..25......1.1..3..9...34.1...
#24
6...8...5.4...128..8.....6...7..23.....5.8.....17......6.....4...43...2.3...9...6
#25
4...1...56..2.7..4......1...4.9...6...2.5.8...9...4.....7......8..5.1..31...6...8
#26
.26...9....5.6.4.....9.7...9..1....5.4....28.5....3..4...6.2.....9.5.7....1...85.
#27
.....1...4..9.8..5..2...6.8.3..5..71.8.....6..7..9..8.3.9...4..6..4.5..7...2.....
#28
..5...8...7.3...1.8..1.5..96......74...2.6...29......83....4..1.1...9.4...6...3..
#29
.....8.6.19.....7...5.2.4..93.....4....2.7....6.....89..3.7.8...5.....26.1.3.....
#30
.5.....6...4.2.3..9...4...5.96...5.....4.8.....5...13.7...6.2.3..2.1.9...3.....7.
#31
....5.....4.2.8.9...7.6.4.2.8.....7.7.6...2.5.5.....4.3.8.2.1...7.9.1.......3....
#32
9...1...3...4.7.....1.5.6...23...49.5.......8..8...72...6.4.5.....1.8...4...9...1
#33
.1.....5.7.8........26.83..9..3.5..7.........8..4.2..6..97.12....3...4...8.....9.

Wonder if they had to restore the site ... or have withdrawn the puzzles.
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re: Michael Mepham's "unsolvable" #23-33

Postby Pat » Thu Sep 21, 2006 1:03 pm

#23-33 were available at
www.sudoku.Org.UK/bifurcation.htm { broken link }
and have now vanished from there;

they are { were -- at the time of the post } still available at the following link --
www.sudoku.Org.UK/unsolvablesSep06.txt { broken link }

Michael Mepham wrote:

    ...8.21...6..4..5.1......72..59.67......8......13.48..25......1.1..3..9...34.1...
    6...8...5.4...128..8.....6...7..23.....5.8.....17......6.....4...43...2.3...9...6
    4...1...56..2.7..4......1...4.9...6...2.5.8...9...4.....7......8..5.1..31...6...8
    .26...9....5.6.4.....9.7...9..1....5.4....28.5....3..4...6.2.....9.5.7....1...85.
    .....1...4..9.8..5..2...6.8.3..5..71.8.....6..7..9..8.3.9...4..6..4.5..7...2.....
    ..5...8...7.3...1.8..1.5..96......74...2.6...29......83....4..1.1...9.4...6...3..
    .....8.6.19.....7...5.2.4..93.....4....2.7....6.....89..3.7.8...5.....26.1.3.....
    .5.....6...4.2.3..9...4...5.96...5.....4.8.....5...13.7...6.2.3..2.1.9...3.....7.
    ....5.....4.2.8.9...7.6.4.2.8.....7.7.6...2.5.5.....4.3.8.2.1...7.9.1.......3....
    9...1...3...4.7.....1.5.6...23...49.5.......8..8...72...6.4.5.....1.8...4...9...1
    .1.....5.7.8........26.83..9..3.5..7.........8..4.2..6..97.12....3...4...8.....9.
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Re: Unsolvable #33

Postby ravel » Thu Sep 21, 2006 1:51 pm

As expected, none of the puzzles needed more than 3 brute force steps (i.e. contradiction chains). The hardest seems to be the last one (#33), where my program only found one way for 3 steps, the first 2 being very long - r1c7<>6 or r2c2<>6 and r3c8<>7. Then the puzzle solves with UR1 and BUG+1.

gurth wrote:When I use a forcing net, starting with a candidate x such as 5e4 and proceeding by SSTS to a contradiction, I am not going to list the SSTS steps, I am merely going to state " ? 5e4, (SSTS), ?? -5e4."
Applying this notation, a solution could be written as
1. ? a7, (SSTS), ?? -a7
2. ? c8, (SSTS), ?? -c8
3. ? a1, (SSTS), ?? -a1

Looks good, but to write out the first 2 steps would become very long and complicated, so i would prefer another solution with shorter steps.
What is *not* customarily done (nor ever, as far as I have seen!) is to *eliminate* a candidate in some cell and then prove a contradiction, thus allowing *placement* of the eliminated candidate !!!

Why is it not customarily done? For no good reason. Simply because nobody has thought of doing it.

This is not true, you can find it in many published solutions in this forum by Carcul and others. See here for a recent sample, where Carcul solved a puzzle by pointing out a contradiction when even 2 candidates are eliminated.
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Postby Carcul » Thu Sep 21, 2006 3:02 pm

Gurth wrote:What is *not* customarily done (nor ever, as far as I have seen!) is to *eliminate* a candidate in some cell and then prove a contradiction, thus allowing *placement* of the eliminated candidate!!!


As Ravel pointed, I have been doing that in the past 7 months. However, its generally much more harder to prove that a candidate must be in a given cell than to prove that a candidate cannot be in a cell, specially in very hard puzzles, or if you are trying to solve a puzzle in a single step. For example, in the hardest puzzles from Henk Collection, if you want to solve them in a single step you almost certainly have to prove that a certain candidate must be in a given cell, not the other way around, and that is often harder than solving the puzzle. Unfortunately, I think that is the main path to reach magic cells.

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Postby gsf » Thu Sep 21, 2006 3:37 pm

bob hanson mentioned this in his hypothesis/(dis)proof posts and documentation circa dec 2005
solver documentation
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Unsolvables #27 and #28

Postby gurth » Fri Sep 22, 2006 8:23 am

Unsolvable #27 solved by Forcing Nets

(1) SSTS.

(2) ? 6k9, SSTS, ?? -6k9, 6g9, SSTS.

(3) ? 7a1, SSTS, ?? -7a1.

(4) ? 7k3, SSTS, ?? -7k3, 7k1, Singles to End.



Unsolvable #28 solved by finding a Ruby

(1) (GET) ? -2b3, SSTS, ?? 2b3. Then, but for one naked pair, it's singles all the way! A magnificent Ruby!

I first defined the concept "Ruby" in GeneralPuzzles - G910,A Pearl. In brief, a Ruby is a cell, at the start, containing more than 2 candidates, where the elimination of one candidate (GET) leads to a contradiction, thus placing that candidate, which in turn leads to immediate solution of the puzzle by simple techniques.

The value of the Ruby being proportionate to
(1) The number of candidates in the Ruby Cell, and
(2) The SIMPLICITY of the techniques needed to solve the puzzle once the Ruby Candidate is placed. Singles only being obviously the best.

Whereas pearls make a problem harder, rubies make it easier - if you can find them ! But they ARE things of beauty, so composers should be generous with them.
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GET and GMET

Postby gurth » Sat Sep 23, 2006 7:14 am

GET and GMET:

My remarks have been quoted out of context. Both in my Prologue and in my exposition of GET, I specifically set my definition within the confines of Forcing Net techniques.

"When I use a forcing net", I said. Later, I said : "As usual, I concentrate on developing the technique of Forcing Nets. HERE I have one novel technique to offer !". (I have taken the liberty of capitalising the relevant word.)

And finally, I said :

"When "bifurcating" to start a Forcing Net, it is customary to *place* a candidate in some cell and then prove a contradiction, thus allowing elimination of the placed candidate.

What is *not* customarily done (nor ever, as far as I have seen!) is to *eliminate* a candidate in some cell and then prove a contradiction, thus allowing *placement* of the eliminated candidate !!!

Why is it not customarily done? For no good reason. Simply because nobody has thought of doing it."

Quite clearly, I was referring to "customary" practices in FORCING NETS.

So it is beside the point to offer me examples of candidate elimination in the domains of AIC, Forcing Chains and Nice Loops. I am well aware that in AIC, the majority of chains in fact start with a strong inference, i.e. the assumption of a false candidate.

I am trying to develop and encourage the use of Forcing Nets, which seem to be avoided and eschewed by Carcul.

I regard a Forcing Chain, or Nice Loop, as a one-dimensional, limited, blinkered version of a Forcing Net. It restricts itself arbitrarily as to what deductions "may" be made, and so ends up much less powerful, and much more difficult to spot and use, than a Forcing Net.

Which I regard as making a virtue of inefficiency.
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Unsolvable #29, with GARNET

Postby gurth » Sat Sep 23, 2006 7:18 am

Unsolvable #29 solved by Forcing Nets, including GET :

(1) SSTS.

(2) ? -2k6, SINGLES ONLY, ?? 2k6. (Here I was trying for a magnificent Ruby, but it was not to be.)

(3) ? 7c2, SSTS, ?? -7c2, 8c2, SSTS.

(4) ? -3a9, SINGLES ONLY, ?? 3a9, SINGLES ONLY TO SOLUTION.
This step, had it occurred at the start, would have been a fantastic Ruby.

Although it is not a Ruby, as it does not occur at the start, still I think it is remarkable and pleasing enough to deserve a name. So I name it, in my hierarchy of jewels, a GARNET.

FORMAL DEFINITION :

A GARNET is in all respects like a Ruby, except that it need not occur immediately, at the start of a puzzle.
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Re: GET and GMET

Postby Myth Jellies » Sat Sep 23, 2006 9:23 am

gurth wrote:I am well aware that in AIC, the majority of chains in fact start with a strong inference, i.e. the assumption of a false candidate.

This assumption absolutely not true.

An AIC does have strong inferences on the endpoints.

The preferred way of defining a strong inference between two candidates is to note that at least one of the candidates must be true.

Of course, an equivalent way of saying that is: for a strong inference to exist between A and B, the implication, "If (not A) then B" must hold true.

However, one can note that a strong implication is valid without ever assuming any of the candidates involved is either true or false. One only has to note that the options given in the premises span the gamut of all possible options. When I have a cell containing only the candidates 'a' and 'b', then I can safely write "a = b" (i.e. 'a' is strongly linked to 'b') without ever assuming that either 'a' or 'b' was false.

AIC's make deductions based upon the relationship of these strong and weak inferences to each other and never assumes anything about any of the candidates or candidate premises. Bifurcation is not involved at all.
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Re: GET and GMET

Postby ravel » Sat Sep 23, 2006 9:25 am

gurth wrote:Quite clearly, I was referring to "customary" practices in FORCING NETS.

So it is beside the point to offer me examples of candidate elimination in the domains of AIC, Forcing Chains and Nice Loops.

Hi Gurth,

we should not mix (the logic behind) techniques and notations. We now, that (regarding the proof for an elimination or placement of candidates) all AIC's and nice loops can be expressed with chains also and vice versa each forcing net can be expressed as nice loop or - i believe - as AIC ( i once did it for a complicated one).
Each notation has its advantages and drawbacks.

Of course a technique that already was used and denoted in NLN or AIC, cannot be claimed to be new, when formulated in forcing nets.
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Puzzle 33: another solution

Postby Carcul » Sat Sep 23, 2006 11:35 am

Code: Select all
 *------------------------------------------------------------------------------*
 | 346      1        46    | 29       23479    3479 | 6789     5        2489    |
 | 7        34569    8     | 1259     123459   349  | 169      1246     1249    |
 | 45       459      2     | 6        14579    8    | 3        147      149     |
 |-------------------------+------------------------+---------------------------|
 | 9        246      146   | 3        168      5    | 18       1248     7       |
 | 123456   234567   14567 | 189      16789    679  | 1589     12348    1234589 |
 | 8        357      157   | 4        179      2    | 159      13       6       |
 |-------------------------+------------------------+---------------------------|
 | 456      456      9     | 7        34568    1    | 2        368      358     |
 | 1256     2567     3     | 2589     25689    69   | 4        1678     158     |
 | 12456    8        14567 | 25       23456    346  | 1567     9        135     |
 *------------------------------------------------------------------------------*

1. [r9c7]=7=[r1c7]=8=[r1c9]-8-[r789c9]-1,5-[r9c7], => r9c7<>1,5.

2. [r5c9]-1,3-[r4c7|r6c8]-8-[r1c7]=8=[r1c9]-8-[r789c9]-1,3-[r5c9], => r5c9<>1,3.

3. [r5c1](-1-[r5c4]=1=[r2c4]-1-[r3c5])(-1-[r6c3])-1-[r8c1]=1=[X-Wing:
r38c89]-1-[r6c8]-3-[r6c23]-5,7-[r145c3]-1-[r5c1], => r5c1<>1.

4. Consider the following sets of cells:
A = r19c3
A' = r149c3
B = r56c3
C = r4c7/r6c8
D = r56c7
E = r4c57
F = r58c6
G = r1c136

Consider also T = r1c7 as our target cell. Ok. Let's now suppose that r9c3 is not 5,7. Then:

-5,7-[r9c3]{=7=[r9c7](-7-[r8c8])-7-[T]}{[A locked on 4,6]-[r4c3=1]-
-[E locked on 6,8]-6-[r5c6]}-[A' locked on 1,4,6]-[B locked on 5,7]-
-[r6c2=3]-[C locked on 1,8](-8-[T])([D locked on 5,9]-9-[T])(-1-[r3c8])-
-1-[r8c8]=1=[r89c9]-1-[r3c9]=1=[r3c5]-1-[r2c4]=1=[r5c4]=8=[r8c4]-8-
-[r8c8]-6-[r8c6]-[F locked on 7,9]-[G locked on 3,4,6]-6-[T]

So, r9c3<>4,6.

5. [r7c8]-6-[r7c12]-5-[r9c3]-7-[r9c7]-6-[r7c8], => r7c8<>6.

6. [r1c9]=8=[r1c7]=7=[r9c7]-7-[r9c3]-5-[r9c4]-2-[r1c4]-9-[r1c79],
=> r1c7/r1c9<>9.

7. Consider now T = r5c4:

[r9c6]{-6-[r58c6](-7-[r5c3])-7,9-[r1c136]-6-[r1c7]}-6-[r9c7]{-7-[r9c3]
(=7=[r6c3])-5-[r9c4]-2-[r1c4]-9-[T]}-7-[r1c7]-8-[r4c7](-1-[r6c78]=1=
=[r6c5]-1-[T])=8=[r4c5]-8-[T], => r9c6<>6.

8. [r2c7](-1-[r2c4]=1=[r5c4]-1-[r4c5])-1-[r4c7]-8-[r4c5]-6-[r5c6]=6=
=[r8c6]-6-[r8c8]-7-[r3c8]-1-[r2c7], => r2c7<>1.

9. [r9c7]=7=[r1c7]-7-[r1c6]-9-[r1c4]-2-[r9c4]-5-[r9c3]-7-[r9c7],
=> r1c5<>7,9; r2c4/r8c4<>2; r9c1/r9c5/r9c9<>5.

10. [r3c2]=9=[r3c9]-9-[r2c7]-6-[r9c7]-7-[r9c3]=7=[r8c2]-7-[r6c2]-5-
-[r3c2], => r3c2<>5.

11. =6=[r5c6](-6-[r4c5])=7=[r1c6]{-7-[r2c4|r3c5](-1,5-[r2c5])-1-[r4c5]
=1=[r4c7]-1-[r6c7]}-7-[r1c7]=7=[r9c7]-7-[r9c3]{-5-[r7c12|r9c4]-2,4,6-
-[r9c1]-1-[r9c9]-3-[r7c9]=3=[r7c5]-3-[r12c5]=3=[r2c6]-3-[r2c2]}-5-
-[r5c3]=5=[r5c7]-5-[r6c7]-9-[r2c7](-6-[r2c2])-6-[r1c7]-8-[r1c9]-{TIUR},
=> r5c6<>6.

12. Now we have r4c3=4 or r5c3=4, and r7c1=4 or r4c2/r5c2=4. So, r7c1=4 and that solves the puzzle.

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Postby Mike Barker » Sun Sep 24, 2006 6:14 am

The Sudoku Unification Method was developed a week ago here. Basically it defines a set of Restricted Subsets (RS - cells, houses, fish, URs, BUGs, ALS, etc) and uses as set of linking elements (direct replacement, direct links, bivalues, stong links, grouped strong links, and ALS) to connect the RS to a Candidate Exclusion Cells (CEC) such that a candidate in the CEC must be false in order to avoid a contradiction condition (CC) in the RS (an empty cell, multivalued cell, incomplete house, incomplete ALS, incomplete covering set, etc.) The SUM approach predicts several new solving techinques. I've implemented several of them. The simplest is the Kraken Blossom which uses a cell as the RS with the no valid candidates in the cell as the CC. This is similar to Death Blossom except that strong links are used in addition to ALS. The Kraken House uses a row, column, or box as the RS and an incomplete house as the contradiction condition. When used with bivalues and strong links these can be used effectively to solve intermediate puzzles (see here). I have also updated the Kraken Fish to use the SUM approach (RS=fish=a digit in n rows or columns, CC=incomplete covering set). In addition, I've added a Kraken Unique Rectangle (RS=UR, CC=deadly pattern).

More advanced problems can be addressed using ALS as a linking element. With this addition, U#25 was solved with a Kraken Cell (Death Blossom) and U#27 with a Kraken Fish. In the examples to date, there has been a single link between the RS and the CEC. In fact, the link between them can be larger. By allowing two links between the RS and the CEC, U#24 can be solved. Here is one solution. There is lots more work required to find the simplest SUM approach, but this at least verifies its effectiveness to address more difficult problems in a logical manner.

Since I first published this solution, I've expanded fish to include Endofins (Obi-Wahns extension to allow fish where two of the base sets include a common cell), so unlike the previous solution this one includes an Endofin Mutant Swordfish. Its not necessary, but for those interested in different solving techniques kind of a fun elimination. The real work horse here is the Kraken Blossom.
    1) Naked Single (40)
    2) Hidden Single (17)
    3) Locked Line/Box (6)
    4) Locked Box/Box (3)
    5) 4-valued/2-link Kraken Unit (3)
    6) Naked Pair (2)
    7) UR+2(X,D,B)/1SL (Type 4,...) (2)
    8) Nice Loops with 3 Strong Links/BV Cells (2)
    9) Grouped Nice Loops with 4 GSL/ALS (2)
    10) A=2 cell ALS-xz rule (2)
    11) Finned X-wing (1)
    12) Finned Swordfish (1)
    13) Mutant Swordfish (1)
    14) UR+3(X,C,N,U,E)/2SL (1)
    15) UR+4(x,X)/2SL, UR+4(x,X)/1SL (1)
    16) Advanced BUG-Lite with 2 lines (1)
    17) Advanced Colouring with 6 Links (1)
    18) Nice Loops with 4 Strong Links/BV Cells (1)
    19) Nice Loops with 5 Strong Links/BV Cells (1)
    20) Strong Nice Loops with 5 GSL/BV Cells (1)
    21) Grouped Nice Loops with 3 GSL/ALS (1)
    22) B=2 cell ALS-xy rule (1)
    23) 3-valued/2-link Kraken Blossom (1)
    24) 4-valued/2-link Kraken Blossom (1)
Code: Select all
Locked Column Line/Box: r46c1 => r78c1<>8
Locked Column Line/Box: r46c8 => r9c8<>5
Locked Column Line/Box: r46c9 => r8c9<>8
Locked Column Box/Box: r137c7|r238c9 => r5c7<>9,r456c9<>9
UR+2D/1SL (5,9): r46c28 => r6c2<>5
Column Finned Swordfish: r2379c3|r379c6|r79c7 => r3c1<>5
3-element Nice Loop: r6c8=5=r6c1-5-r4c2-9-r4c4=9=r6c6~9~r6c8 => r6c8<>9
5-element Strong Nice Loop: r4c2-9-r4c4=9=r6c6=3=r3c6=5=r23c5-5-r8c5=5=r8c12~5~ => r9c2<>5
A=2 cell ALS xz-rule: r1c34-9-r179c6 => r3c6<>4
A=2 cell ALS xz-rule: r56c5-1-r8c5|r79c6 => r6c6<>4
B=2 cell ALS xy-rule: r179c6-9-r13c4-2-r234568c5 => r7c5<>5
Overlap 4-element Grouped Nice Loop: r1c3-9-ALS:r179c6-5-r8c5=5=r8c12-5-ALS:r179c3~2~ => r3c3<>2
+----------------------+--------------------+------------------+
|     6   1279    29*d |  249      8  479*b | 1479    3     5  |
|   579      4    359  |   69   3567     1  |    2    8    79  |
|  1279      8  359-2  |  249  23457  3579  | 1479    6  1479  |
+----------------------+--------------------+------------------+
|  4589     59      7  | 1469    146     2  |    3   19   148  |
|   249    239      6  |    5    134     8  |  147  179  1247  |
|  2489    239      1  |    7     34    39  |    6    5   248  |
+----------------------+--------------------+------------------+
| 12579      6  2589*d |  128    127   57*b | 1579    4     3  |
| 1579*c 1579*c     4  |    3    157*    6  |    8    2   179  |
|     3    127   258*d | 1248      9  457*b |  157   17     6  |
+----------------------+--------------------+------------------+
4-element Grouped Nice Loop: ALS:r2c19-5-r23c3=5=r79c3-5-r8c12=5=r8c5-5-ALS:r179c6~9~ => r2c4<>9
+----------------------+--------------------+------------------+
|     6   1279     29  |  249      8  479*e | 1479    3     5  |
|   579*     4   359*b |  6-9   3567     1  |    2    8    79* |
|  1279      8   359*b |  249  23457  3579  | 1479    6  1479  |
+----------------------+--------------------+------------------+
|  4589     59      7  | 1469    146     2  |    3   19   148  |
|   249    239      6  |    5    134     8  |  147  179  1247  |
|  2489    239      1  |    7     34    39  |    6    5   248  |
+----------------------+--------------------+------------------+
| 12579      6  2589*c |  128    127   57*e | 1579    4     3  |
| 1579*d 1579*d     4  |    3    157*    6  |    8    2   179  |
|     3    127   258*c | 1248      9  457*e |  157   17     6  |
+----------------------+--------------------+------------------+
Endofin Mutant Swordfish (r28c3/c9b17): r2c139|r8c129|r137c3 => r3c1<>9,r1c2<>9
+---------------------+--------------------+------------------+
|     6  127-9    29* |  249      8   479  | 1479    3     5  |
|   579*     4   359# |    6    357     1  |    2    8    79* |
| 127-9      8   359* |  249  23457  3579  | 1479    6  1479  |
+---------------------+--------------------+------------------+
|  4589     59     7  |  149      6     2  |    3   19   148  |
|   249    239     6  |    5    134     8  |  147  179  1247  |
|  2489    239     1  |    7     34    39  |    6    5   248  |
+---------------------+--------------------+------------------+
| 12579      6  2589* |  128    127    57  | 1579    4     3  |
|  1579*  1579*    4  |    3    157     6  |    8    2   179* |
|     3    127   258  | 1248      9   457  |  157   17     6  |
+---------------------+--------------------+------------------+
3-element Grouped Nice Loop: ALS:r4c48-4-r56c5=4=r3c5-4-ALS:r238c9~1~ => r4c9<>1
+--------------------+--------------------+-------------------+
|     6   127    29  |  249      8   479  | 1479    3      5  |
|   579     4   359  |    6    357     1  |    2    8    79*c |
|   127     8   359  |  249  23457* 3579  | 1479    6  1479*c |
+--------------------+--------------------+-------------------+
|  4589    59     7  |  149*     6     2  |    3   19*  48-1  |
|   249   239     6  |    5   134*b    8  |  147  179   1247  |
|  2489   239     1  |    7    34*b   39  |    6    5    248  |
+--------------------+--------------------+-------------------+
| 12579     6  2589  |  128    127    57  | 1579    4      3  |
|  1579  1579     4  |    3    157     6  |    8    2   179*c |
|     3   127   258  | 1248      9   457  |  157   17      6  |
+--------------------+--------------------+-------------------+
UR+3C/2SL (4,8): r46c19 => r6c1<>4
UR+4X/2SL (8,2): r79c34, r9c2678 => r7c3<>2
4-valued/2-element Kraken Row (r1c2=1=r1c7-1-, r1c6-7-r7c6-5-r7c7|r9c8|r8c9-1-, r1c7-7-r2c9-9-r9c8|r8c9-1-): r1c267=7 => r9c7<>1
+-------------------+--------------------+-------------------+
|     6  127*b  29  |  249      8   479* | 1479*b   3     5  |
|   579     4  359  |    6    357     1  |     2    8    79d |
|   127     8  359  |  249  23457  3579  |  1479    6  1479  |
+-------------------+--------------------+-------------------+
|  4589    59    7  |  149      6     2  |     3   19    48  |
|   249   239    6  |    5    134     8  |   147  179  1247  |
|   289   239    1  |    7     34    39  |     6    5   248  |
+-------------------+--------------------+-------------------+
| 12579     6  589  |  128    127    57c |  1579c   4     3  |
|  1579  1579    4  |    3    157     6  |     8    2  179cd |
|     3   127  258  | 1248      9   457  |  57-1  17cd    6  |
+-------------------+--------------------+-------------------+
UR+2D/1SL (5,7): r79c67 => r9c6<>5
6-element Advanced Colouring: r9c7=5=r7c7-5-r7c6=5=r3c6=3=r6c6=9=r4c4-9-r4c8=9=r5c8=7=r9c8~7~r9c7 => r9c7<>7
4-valued/2-element Kraken Row (r7c1-7-r23c1=7=r1c2-7-r13c4|r1c6-9-, r7c5=2=r3c5-2-r13c4-9-, r7c6-7-r19c6-9-, r7c7-7-r49c8-9-r4c4=9=r6c6-9-): r7c1567=7 => r3c6<>9
+-------------------+---------------------+------------------+
|     6   127b  29  | 249bc     8   479bd | 1479    3     5  |
|   579b    4  359  |    6    357      1  |    2    8    79  |
|   127b    8  359  | 249bc 23457c 357-9  | 1479    6  1479  |
+-------------------+---------------------+------------------+
|  4589    59    7  |  149e     6      2  |    3   19e   48  |
|   249   239    6  |    5    134      8  |  147  179  1247  |
|   289   239    1  |    7     34     39e |    6    5   248  |
+-------------------+---------------------+------------------+
| 12579*    6  589  |  128   127*c    57* |  179*   4     3  |
|  1579  1579    4  |    3    157      6  |    8    2   179  |
|     3   127   28  | 1248      9     47d |    5   17e    6  |
+-------------------+---------------------+------------------+
3-valued/2-element Kraken Blossom (r9c4-1-r9c28-2-r6c26-9-, r9c4-2-r13c4-9-r4c4=9=r6c6-9-, r9c4-4-r13c4-9-r4c4=9=r6c6-9-, r9c4-8-r19c3-9-r1c6=9=r6c6-9-): r9c4=1248 => r6c1<>9
+-------------------+---------------------+------------------+
|     6   127   29e | 249cd     8    479e | 1479    3     5  |
|   579     4  359  |    6    357      1  |    2    8    79  |
|   127     8  359  | 249cd 23457    357  | 1479    6  1479  |
+-------------------+---------------------+------------------+
|  4589    59    7  | 149cd     6      2  |    3   19    48  |
|   249   239    6  |    5    134      8  |  147  179  1247  |
|  28-9   239b   1  |    7     34  39bcde |    6    5   248  |
+-------------------+---------------------+------------------+
| 12579     6  589  |  128    127     57  |  179    4     3  |
|  1579  1579    4  |    3    157      6  |    8    2   179  |
|     3   127b  28e | 1248*     9     47  |    5   17b    6  |
+-------------------+---------------------+------------------+
Advanced 2-line BUG Lite (XY:r6c1|r4c9-8-r6c9, SL:r4c1=4=r5c1, SL:r6c9=2=r5c9): r456c19 => r4c1<>8,r6c9<>8
4-valued/2-element Kraken Row (r9c2-1-r1c2=1=r1c7=4=r1c46-4-, r9c4-1-r9c28-2-r6c256-4-, r9c8-1-r4c8=1=r4c4=4=r56c5-4-): r9c248=1 => r3c5<>4
+-------------------+--------------------+------------------+
|     6   127b  29  |  249b      8  479b | 1479b   3     5  |
|   579     4  359  |    6     357    1  |    2    8    79  |
|   127     8  359  |  249  2357-4  357  | 1479    6  1479  |
+-------------------+--------------------+------------------+
|   459    59    7  |  149d      6    2  |    3   19d    8  |
|   249   239    6  |    5     134d   8  |  147  179  1247  |
|     8   239c   1  |    7     34cd  39c |    6    5    24  |
+-------------------+--------------------+------------------+
| 12579     6  589  |  128     127   57  |  179    4     3  |
|  1579  1579    4  |    3     157    6  |    8    2   179  |
|     3  127*c  28  | 1248*      9   47  |    5  17*c    6  |
+-------------------+--------------------+------------------+
Locked Column Box/Box: r139c4|r19c6 => r4c4<>4
5-element Nice Loop: r7c4=8=r9c4-8-r9c3-2-r1c3-9-r1c6=9=r6c6-9-r4c4~1~r7c4 => r7c4<>1
3-element Nice Loop: r7c4-8-r7c3=8=r9c3=2=r1c3~2~ => r1c4<>2
Locked Row Line/Box: r3c45 => r3c1<>2
4-valued/2-element Kraken Blossom (r7c1-1-r3c1=1=r1c2-1-r1c467-9-, r7c1-2-r7c4-8-r237c3-9-, r7c1-5-r7c6-7-r1c46-9-, r7c1-7-r23c1=7=r1c2-7-r1c46-9-, r7c1-9-r7c7=9=r8c9-9-r2c9=9=r2c13-9-): r7c1=12579 => r1c3<>9
+--------------------+--------------------+------------------+
|     6  127be  2-9  | 49bde    8  479bde | 1479b   3     5  |
|   579e    4  359ce |    6   357      1  |    2    8    79e |
|   17be    8   359c |  249  2357    357  | 1479    6  1479  |
+--------------------+--------------------+------------------+
|     4     5     7  |   19     6      2  |    3   19     8  |
|    29   239     6  |    5   134      8  |  147  179  1247  |
|     8   239     1  |    7    34     39  |    6    5    24  |
+--------------------+--------------------+------------------+
| 12579*    6   589c |   28c  127     57d |  179e   4     3  |
|  1579   179     4  |    3   157      6  |    8    2   179e |
|     3   127    28  | 1248     9     47  |    5   17     6  |
+--------------------+--------------------+------------------+
Naked Block Pair: r3c1|r1c2 => r2c1<>7
4-element Nice Loop: r7c3-5-r7c6=5=r3c6=3=r6c6=9=r6c2~9~ => r8c2<>9
Locked Column Box/Box: r278c1|r237c3 => r5c1<>9
Locked Row Line/Box: r1c46 => r1c7<>4
Locked Row Line/Box: r1c46 => r1c7<>9
Naked Row Pair: r1c27 => r1c6<>7
Row Finned X-Wing: r3c179|r7c17 => r1c7<>1
Last edited by Mike Barker on Sat Jan 26, 2008 1:05 pm, edited 2 times in total.
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Postby daj95376 » Sun Sep 24, 2006 8:08 am

Mike Barker, I admit that I'm way out of my league here, but I do have a question and a comment even though I've only examined part of your solution for U#24.

Q: In your second SUM elimination, did you drop the details for r8c1-9-... because they were a mirror of r8c2 and would take up unnecessary space?

[Updated:]
C: The first two SUM eliminations remind me of complex Triple Implication Chains. I guess this is one of the techniques SUM includes in its scope.
Last edited by daj95376 on Sun Sep 24, 2006 8:50 am, edited 1 time in total.
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Unsolvable #30

Postby gurth » Sun Sep 24, 2006 9:44 am

Unsolvable #30 solved by Forcing Nets and fine GARNET :

1. SSTS.

2. ? -4d8, SSTS, ?? 4d8, SSTS.

3. ? 4k7, SSTS, ?? -4k7, SSTS.

4. ? -4k1, SSTS, ?? -4k1.

5. ? 4h2, 6h9, 8k7, 5h8, 1g8, 8g2h1 ?? -4h2.

6. ? -4g2, 4g6, 4k9, 6h9, 8k7, 5h8, 1g8, 8gh2 ?? 4g2, SSTS.

7. ? -8c3, SINGLES ONLY, ?? 8c3, SINGLES TO END.

Step 7 completes the solution with a very fine GARNET, the trivalue cell at c3, requiring singles only for both contradiction of -8c3 and subsequent completion of the puzzle.
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