The New Sudoku Players' Forum

[udosuk]

This is the original puzzle:

Code: Select all

..4|3.8|.1.
...|.94|8..
9..|1..|.4.
---+---+---
5..|.16|9..
1..|...|..4
..8|74.|..5
---+---+---
.1.|..5|..2
..7|98.|...
.6.|4.1|7..

After singles/locked candidates/pairs:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 | 236   235   1     | 25    9     4     | 8     2357  367   |
 | 9     8     2356  | 1     25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 | 236   9     8     | 7     4     23    | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 348   1     39    | 6     7     5     | 34    389   2     |
 | 234   235   7     | 9     8     23    | 3456  356   1     |
 | 238   6     59    | 4     23    1     | 7     59    38    |
 *-----------------------------------------------------------*

Here SS will apply swordfish, simple colors, multiple colors and xy-wings to solve it...

However, I'm seeking the shortest route to solve it... Shortest meaning the fewest advanced moves required...

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 |-236   235   1     |#25    9     4     | 8     2357  367   |
 | 9     8     2356  | 1    #25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 |-236   9     8     | 7     4    @23    | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 348   1     39    | 6     7     5     | 34    389   2     |
 | 234   235   7     | 9     8    @23    | 3456  356   1     |
 |*238   6     59    | 4    *23    1     | 7     59    38    |
 *-----------------------------------------------------------*

Firstly, 2 turbots (simple colors) eliminate 2 from r26c1:
[*+@]: r6c1-2-r6c6=2=r8c6=2=r9c5=2=r9c1-2-r6c1 => r6c1<>2
[*+#]: r2c1-2-r2c4=2=r3c5-2-r9c5=2=r9c1-2-r2c1 => r2c1<>2

These lead to some locked candidates in c123, plus a naked pair {36} in r26c1, which gives us a swordfish on 3:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 |*36   *235   1     | 25    9     4     | 8    -2357 -367   |
 | 9     8     356   | 1     25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 |*36    9     8     | 7     4    *23    | 1    -236   5     |
 |-------------------+-------------------+-------------------|
 | 48    1     39    | 6     7     5     | 34    389   2     |
 | 24   *35    7     | 9     8    *23    |-3456 -356   1     |
 | 28    6     59    | 4     23    1     | 7     59    38    |
 *-----------------------------------------------------------*

Column swordfish on 3 in r268c126 => 3s eliminated from r268c345789

Next, a finned sashimi jellyfish on 2 enables us to eliminate 2 from r5c7:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 | 36    235   1     | 25    9     4     | 8     257   67    |
 | 9     8     356   | 1    *25    7     |*235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     |-236   2368  4     |
 | 36    9     8     | 7     4    *23    | 1    #26    5     |
 |-------------------+-------------------+-------------------|
 | 48    1     39    | 6     7     5     | 34    389   2     |
 |*24    35    7     | 9     8    *23    | 456   56    1     |
 |*28    6     59    | 4    *23    1     | 7     59    38    |
 *-----------------------------------------------------------*

Row finned sashimi jellyfish on 2 in r3689c1567 with fin in b6 (r6c8) => r5c7<>2

(SS uses a multiple colors move which doesn't involve r89c1, equivalent to a "mutant finned sashimi swordfish")

Eventually we find an xy-wing to eliminate the 3 in r5c5:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 | 36    235   1     | 25    9     4     | 8     257   67    |
 | 9     8     356   | 1     25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258  -235   9     |*36    2368  4     |
 | 36    9     8     | 7     4    *23    | 1    *26    5     |
 |-------------------+-------------------+-------------------|
 | 48    1     39    | 6     7     5     | 34    389   2     |
 | 24    35    7     | 9     8     23    | 456   56    1     |
 | 28    6     59    | 4     23    1     | 7     59    38    |
 *-----------------------------------------------------------*

r6c8=2|6 => either r5c7 or r6c6 (or both) must be 3 => r5c5<>3

The puzzle is then solved with singles...

My question is, can any of the 5 major steps above be replaced/combined?

I don't like complex forcing chains, but xy-wings/xyz-wings/xyzw-wings are okay, and uniqueness moves are alright too...

As an example, the first 2 turbots perhaps could be replaced:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 |*236   235   1     |*25    9     4     | 8     2357  367   |
 | 9     8     2356  | 1    *25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 |*236   9     8     | 7     4    *23    | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 348   1     39    | 6     7     5     | 34    389   2     |
 |*234  -235   7     | 9     8    *23    | 3456  356   1     |
 |*238   6     59    | 4    *23    1     | 7     59    38    |
 *-----------------------------------------------------------*

In b8, either r8c6=2 or r9c5=2
r8c6=2 => r8c2<>2
r9c5=2 => r8c6,r3c5<>2 => r6c6=r2c4=2 => r26c1<>2 => r8c1=2 => r8c2<>2

Therefore r8c2<>2

From there we can find a hidden triple {248} in r789c1 and proceed with the ensuring swordfish on 3...

However, I couldn't find an elegant fish to represent the above reasoning... Any fishing expert?

I suspect the solving route (without complex forcing chains) must follow the routine:

1. eliminate 2 from r26c1 or r8c2 (using turbots/simple colors)
2. eliminate 3 from r789c1 (using a hidden triple/naked pair)
3. eliminate 3 from r6c8 (using a swordfish)
4. eliminate 2 from r5c7 (using a finned sashimi jellyfish/mutant finned sashimi swordfish/multiple colors)
5. eliminate 3 from r5c5 (using an xy-wing)

Is there any detour?

[re'born]

Well, I'm not sure if this is what you are looking for, but here is how I solved it. After the basic techniques, I got to:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 | 236   235   1     | 25d   9     4     | 8     2357  367   |
 | 9     8     2356  | 1     25c   7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28e   1     6     | 9     2378  378-  |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 | 236   9     8     | 7     4     23    | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 348   1     39    | 6     7     5     | 34    389   2     |
 | 234   235   7     | 9     8     23    | 3456  356   1     |
 | 238   6     59    | 4     23b   1     | 7     59    38a   |
 *-----------------------------------------------------------*


where an xy-chain 8-[r9c9]-3-[r9c5]-2-[r3c5]-5-[r2c4]-2-[r4c4]-8 eliminates the 8 from r4c9. A few singles later we get

Code: Select all

  *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 | 236   235   1     | 25    9     4     | 8     2357  367   |
 | 9     8     2356  | 1     25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  37    |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 | 236-  9     8     | 7     4     23*   | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 8     1     39    | 6     7     5     | 4     39    2     |
 | 4     235   7     | 9     8     23*   | 356   356   1     |
 | 23*   6     59    | 4     23*   1     | 7     59    8     |
 *-----------------------------------------------------------*


where remote naked pairs implies r6c1=6. It's all singles from here.

[Ruud]

I found a 3-step detour:

Code: Select all

.---------------.---------------.---------------.
| 7    25   4   | 3    6    8   | 25   1    9   |
| 236  235  1   | 25   9    4   | 8    2357 367 |
| 9    8    2356| 1    25   7   | 235  4    36  |
:---------------+---------------+---------------:
| 5    4    23  | 28   1    6   | 9    2378 378 |
| 1    7    236 | 258  235  9   | 236  2368 4   |
| 236  9    8   | 7    4    23  | 1    236  5   |
:---------------+---------------+---------------:
| 348  1   #39  | 6    7    5   |-34  *389  2   |
| 234  235  7   | 9    8    23  | 3456 356  1   |
| 238  6    59  | 4    23   1   | 7    59  #38  |
'---------------'---------------'---------------'
 XYZ-Wing - r7c7=4

.---------------.---------------.---------------.
| 7    25   4   | 3    6    8   | 25   1    9   |
| 236  235  1   | 25   9    4   | 8    2357 367 |
| 9    8    2356| 1    25   7   | 235  4    36  |
:---------------+---------------+---------------:
| 5    4    23  | 28   1    6   | 9    2378 378 |
| 1    7    236 | 258  235  9   | 236  2368 4   |
|-236  9    8   | 7    4   *23  | 1    236  5   |
:---------------+---------------+---------------:
| 38   1    39  | 6    7    5   | 4    389  2   |
| 4    235  7   | 9    8   *23  | 356  356  1   |
|*238  6    59  | 4   *23   1   | 7    59   38  |
'---------------'---------------'---------------'
 Turbot Fish (2-string kite) digit 2

.---------------.---------------.---------------.
| 7    25   4   | 3    6    8   | 25   1    9   |
|-236  235  1   |*25   9    4   | 8    2357 367 |
| 9    8    356 | 1   *25   7   | 235  4    36  |
:---------------+---------------+---------------:
| 5    4    23  | 28   1    6   | 9    2378 378 |
| 1    7    236 | 258  235  9   | 236  2368 4   |
| 36   9    8   | 7    4    23  | 1    236  5   |
:---------------+---------------+---------------:
| 38   1    39  | 6    7    5   | 4    389  2   |
| 4    235  7   | 9    8    23  | 356  356  1   |
|*238  6    59  | 4   *23   1   | 7    59   38  |
'---------------'---------------'---------------'
 Empty Rectangle digit 2

Singles all the way from here.

Ruud

[udosuk]

Thanks for the help guys...

rep'nA, despite your xy-chain involves only 3 candidates and is quite short IMHO it feels less elegant than the more standard pattern moves... But good spotting nonetheless...

Ruud, thanks, your xyz-wing is exactly what I'm looking for... The last empty rectangle coincidentally is the same as a turbot fish, so I'll need to have a look to see the subtle difference in the logic between them...

However, I'm still looking for a single fish to combine those 2 eliminations of 2s in r26c1, or to eliminate the 2 in r8c2...

[daj95376]

However, I'm still looking for a single fish to combine those 2 eliminations of 2s in r26c1, or to eliminate the 2 in r8c2...

OR ... eliminate <2> in [r6c6] or [r9c5].

Either [r8c6]=2 or ( [r6c6]=2 and [r8c2]=2 and [r9c5]=2 ). As it turns out, [r8c6]=2 is true and leads to a cascade of eliminations in <2> followed by Singles to complete the puzzle.

(Edit #1: corrected [r9c4] to be [r9c5])
(Edit #2: I started from Ruud's PM after XYZ-Wing elimination.)

[udosuk]

OR ... eliminate <2> in [r6c6] or [r9c4].

I take it you mean [r9c5] instead of [r9c4]?

Either [r8c6]=2 or ( [r6c6]=2 and [r8c2]=2 and [r9c4]=2 ). As it turns out, [r8c6]=2 is true and leads to a cascade of eliminations in <2> followed by Singles to complete the puzzle.

Either [r8c6]=2 or ( [r6c6]=2 and [r8c1]=2 and [r9c5]=2 )...

I don't know why you wrote [r8c2]=2 above...
And how could you determine [r8c6]=2 was the true case without trial and error?

But thanks for trying to help...

[ronk]

However, I'm seeking the shortest route to solve it... Shortest meaning the fewest advanced moves required...

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 |-236   235   1     |#25    9     4     | 8     2357  367   |
 | 9     8     2356  | 1    #25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 |-236   9     8     | 7     4    @23    | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 348   1     39    | 6     7     5     | 34    389   2     |
 | 234   235   7     | 9     8    @23    | 3456  356   1     |
 |*238   6     59    | 4    *23    1     | 7     59    38    |
 *-----------------------------------------------------------*

Firstly, 2 turbots (simple colors) eliminate 2 from r26c1:
[*+@]: r6c1-2-r6c6=2=r8c6=2=r9c5=2=r9c1-2-r6c1 => r6c1<>2
[*+#]: r2c1-2-r2c4=2=r3c5-2-r9c5=2=r9c1-2-r2c1 => r2c1<>2

However, I'm still looking for a single fish to combine those 2 eliminations of 2s in r26c1, or to eliminate the 2 in r8c2...

There is an ultimate fish for simultaneous eliminations r26c1<>2, but it's certainly not easier than two turbots.

Code: Select all

 7    25   4    | 3    6    8    | 25   1    9
-236  235  1    |*25   9    4    | 8    2357 367
 9    8    2356 | 1   *25   7    | 235  4    36
----------------+----------------+---------------
 5    4    23   | 28   1    6    | 9    2378 378
 1    7    236  | 258  235  9    | 236  2368 4
-236  9    8    | 7    4   *23   | 1    236  5
----------------+----------------+---------------
 348  1    39   | 6    7    5    | 34   389  2
*234 *235  7    | 9    8   *23   | 3456 356  1
@238  6    59   | 4   *23   1    | 7    59   38

 sashimi mutant jellyfish r9c6b27\r268c5 plus endo-fin r9c1, implies r26c1<>2

If fin r9c1 were false, all the candidates in base set r9, c6, b2 and b7 would be covered by cover set r2, r6, r8 and c5. Any candidates in the cover set but not in the base set would be excluded. R26c1<>2 would be two of those exclusions.

If r9c1 were true, r26c1<>2 would still be valid because both exclusion cells see the fin.

Even if r9c1 could be covered, it would still be a fin because it lies in the intersection of the two base units (sectors) r9 and b7. Such a cell has been recently dubbed an endo-fin (endo-fin-cell, to be precise).

I didn't see anything easy for r8c2<>2.

[udosuk]

Thanks heaps for the ultimate fish, Ron!

I think when you go fishing, one huge fish is definitely better than 2 small fish, no matter how difficult it is to catch...

Inspired by your work I think I caught another sashimi mutant jellyfish for the r8c2 elimination:

Code: Select all

 *-----------------------------------------------------------*
 | 7     25    4     | 3     6     8     | 25    1     9     |
 |#236   235   1     |#25    9     4     | 8     2357  367   |
 | 9     8     2356  | 1    #25    7     | 235   4     36    |
 |-------------------+-------------------+-------------------|
 | 5     4     23    | 28    1     6     | 9     2378  378   |
 | 1     7     236   | 258   235   9     | 236   2368  4     |
 |#236   9     8     | 7     4    #23    | 1     236   5     |
 |-------------------+-------------------+-------------------|
 | 348   1     39    | 6     7     5     | 34    389   2     |
 |#234  -235   7     | 9     8    #23    | 3456  356   1     |
 |@238   6     59    | 4    #23    1     | 7     59    38    |
 *-----------------------------------------------------------*
sashimi mutant jellyfish r9c16b2\r268c5 plus endo-fin r9c1 => r8c2<>2

If fin r9c1=2, r8c2 must not be 2 (same box).

If r9c1<>2, all the candidates in base set r9,c1,c6,b2 would be covered by cover set r2,r6,r8,c5.
Any candidate in the cover set but not in the base set would be excluded.
r8c2 would be one of those exclusions.

Hence r8c2<>2

[ronk]

sashimi mutant jellyfish r9c16b2\r268c5 plus endo-fin r9c1 => r8c2<>2
(...)
If fin r9c1=2, r8c2 must not be 2 (same box).

If r9c1<>2, all the candidates in base set r9,c1,c6,b2 would be covered by cover set r2,r6,r8,c5.
Any candidate in the cover set but not in the base set would be excluded.
r8c2 would be one of those exclusions.

Hence r8c2<>2

Excellent catch. I'm embarassed to not have seen that.

I think when you go fishing, one huge fish is definitely better than 2 small fish, no matter how difficult it is to catch...

I'll take "better" to mean "more satisfying" and wholehearedly agree. Sorta like catching two fish on one cast.

[Carcul]

Code: Select all

 *-----------------------------------------------------*
 | 7     25    4    | 3     6     8  | 25    1     9   |
 | 236   235   1    | 25    9     4  | 8     2357  367 |
 | 9     8     2356 | 1     25    7  | 235   4     36  |
 |------------------+----------------+-----------------|
 | 5     4     23   | 28    1     6  | 9     2378  378 |
 | 1     7     236  | 258   235   9  | 236   2368  4   |
 | 236   9     8    | 7     4     23 | 1     236   5   |
 |------------------+----------------+-----------------|
 | 348   1     39   | 6     7     5  | 34    389   2   |
 | 234   235   7    | 9     8     23 | 3456  356   1   |
 | 238   6     59   | 4     23    1  | 7     59    38  |
 *-----------------------------------------------------*


r6c1=6 or r9c1=8. But [r9c9]=8=[r4c9]-8-[r4c4]-2-[r6c6]=2=[r8c6]=3=[r9c5]-3-[r9c9]. So r9c9=8 and the puzzle is solved.

Carcul

[ronk]

r6c1=6 or r9c1=8. But [r9c9]=8=[r4c9]-8-[r4c4]-2-[r6c6]=2=[r8c6]=3=[r9c5]-3-[r9c9]. So r9c9=8 and the puzzle is solved.

So how is knowing that "r6c1=6 or r9c1=8" helpful?

[ravel]

udosuk,

where you have marked the swordfish above, there is a simple xy-wing (r9c1<>8), which solves the puzzle.

[udosuk]

Thanks ravel, I must be blind not to see it again! (You caught me twice a day! )

So 2 advanced steps (the sashimi mutant jellyfish with endo-fin pointed out by ronk plus this xy-wing) solve it... Ruud's detour (the sashimi mutant jellyfish plus an xyz-wing) works fine too...

Carcul, thanks for the forcing chain too, but my taste for elegancy doesn't allow any chains involving 3 or more digits with a length longer than an xy-wing or xyz-wing!

ronk, "r6c1=6 or r9c1=8" is helpful because after you set r9c9=8 you can immediately put 6 in r6c1... Though it is effectively just a "remote pair" move...

[_m_k]

This method uses only elimination and hidden singles!
We have r6c6=2 or 3 and r9c1=2 or 3 or 8.
Claim: r6c6=2 => r9c1<>2 and r9c1<>3 and r9c1<>8, a contradiction.
Hence, r6c6<>2, or r6c6=3 and puzzle is solved using only elimination and hidden singles.
Proof of claim:
r6c6=2 => r8c6=3 => r9c5=2 => r9c1<>2.
r6c6=2 => r3c4=8 => r3c9<>8 => r9c9=8 (hidden single in c9) => r9c1<>8.
So r6c6=2 => r9c1=3. Also
r6c6=2 => r6c1=3 or 6.
r6c6=2 => r8c6=3 => r9c5=2 => r3c5=5 => r2c4=2 =>r2c1=3 or 6.
r9c1=3 and r6c1=3 or 6 and r2c1=3 or 6 is a contradiction in c1, and so
r6c6=2 => r9c1<>3.

P.S. Of course I got this idea from my computer program.

[udosuk]

Thanks _m_k for demonstrating your moves... You might not be aware but what you did would be considered as "trial and error" by many players... Your approach is perfectly logical but is felt somewhat less "elegant" than the other moves shown here (subjected to personal opinions). But thanks for you effort nonetheless!