Welcome!
This is part of JCOWebsite and contains the proposed...
I have been solving Sudoku-X puzzles lately.
The following 6x6 Sudoku-X puzzle has nice features.
(generated and displayed with Sudoku6Explainer)
Update1 (Jan 22, 2025), Update 2 (Jan 30, 2025). After the basics moves,
0.1 (6): a5=c5 => -6 e5, f5
0.2 (5): a6 = b5 => -5 a5, c5, b6
0.3 (3): f1 = e2 => -3 d2, f2, e1
0.4 (5): c2 = c3 => -5 b2
0.5 (3): d4 = d5 => -3 e5
0.6 (2): b2 = f6 => -2 b6, -2 f2
0.7 (2): b5 = c5 => -2 d5, f5
0.8 (2)b5 = (2)f1 => -2 b1,
we get the following configuration (corrected!).
+-------------------+-------------------+
| 15 14 3 | 124 1456 1256 | 6
| 146 125 126 | 134 145 135 | 5
+-------------------+-------------------+
| 2 6 4 | 13 135 135 | 4
| 135 135 15 | 6 2 4 | 3
+-------------------+-------------------+
| 13456 1234 1256 | 124 13 16 | 2
| 1346 134 126 | 5 146 123 | 1
+-------------------+-------------------+
a b c d e f Sudoku-X
The puzzle is solved with singles after
just one move that includes the strong link:
(1)e2.f1 = (1)a6.b5
(Find an ALS W-wing)
I propose the following 6x6 tough Sudoku-X puzzle.
A pond with many fishes, a couple birds (wings) and a monster!
(generated and displayed with Sudoku6Explainer)
Update 1 (Feb 3, 2025)
After basics moves, one should get the following configuration:
+-------------------+-------------------+
| 136 136 36 | 2 4 5 | 6
| 25 25 4 | 3 16 16 | 5
+-------------------+-------------------+
| 13456 1346 25 | 146 356 1236 | 4
| 13456 1346 236 | 146 356 1236 | 3
+-------------------+-------------------+
| 23456 2346 56 | 156 136 1346 | 2
| 346 3456 1 | 56 2 346 | 1
+-------------------+-------------------+
a b c d e f Sudoku-X
This puzzle is solvable in ten steps, but one of them is tough! (the monster)
Step 1: fish on 6 to get -6 f1 (6 is false at cell f1)
Step 2: fish on 4 to get -4 b2 (and locked candidate (LC) elimination -4 a4)
Step 3: A nice fish loop on 4 to get -4 a3, -4 b1
Step 4: in the following configuration
+----------------------+-------------------+
| 136 136 36 | 2 4 5 | 6
| 25 25 4 | 3 16 16 | 5
+----------------------+-------------------+
| 1356 1346 25 | 146 356 1236 | 4
| 1356 1346 236 | 146 356 1236 | 3
+----------------------+-------------------+
| 23456 236 56 | 156 136 1346 | 2
| 346 536 1 | 56 2 34 | 1
+----------------------+-------------------+
a b c d e f Sudoku-X
there is an H-wing that eliminates 6(b2).
Step 5: fish on 5 to get -5 a2.
Step 6: In the following conguration
+-----------------------+---------------------+
| 136 136 36 | 2 4 5 | 6
| 25 25 4 | 3 16 16 | 5
+-----------------------+---------------------+
| 1356 1346 25 | 146 356 1236 | 4
| 1356 1346 236 | 146 356 2316 | 3
+-----------------------+---------------------+
| 2346 23 56 | 156 136 1346 | 2
| 346 356 1 | 56 2 34 | 1
+-----------------------+---------------------+
a b c d e f Sudoku-X
there is an S-wing that eliminates (3)f3.
Step 7: in the following configuration
+-----------------------+---------------------+ | 136 136 36 | 2 4 5 | 6 | 25 25 4 | 3 16 16 | 5 +-----------------------+---------------------+ | 1356 1346 25 | 146 356 1236 | 4 | 1356 1346 236 | 146 356 126 | 3 +-----------------------+---------------------+ | 2346 23 56 | 156 136 1346 | 2 | 346 356 1 | 56 2 34 | 1 +-----------------------+---------------------+ a b c d e f Sudoku-X
there is an H-wing with transport that eliminates (1)f3.
Step 8 (hardest!): in the following configuration
+-------------------+-------------------+ | 136 136 36 | 2 4 5 | 6 | 25 25 4 | 3 16 16 | 5 +-------------------+-------------------+ | 1356 1346 25 | 146 356 1236 | 4 | 1356 @1346 236 | 146 356 26 | 3 +-------------------+-------------------+ | 2346 23 56 | 156 #136 1346 | 2 | 346 356 1 | 56 2 34 | 1 +-------------------+-------------------+ a b c d e f Sudoku-X
there is a double kraken at (1346)b3 and (136)e2
that eliminates (2)a2, and this gives 7 placements
and three LC eliminations: -6 a2, -3 a3, -3 a4.
(more on how to find this move at the end)
Step 9: in the following configuration
+----------------+----------------+ | 136 136 36 | 2 4 5 | 6 | 2 5 4 | 3 16 16 | 5 +----------------+----------------+ | 156 1346 2 | 146 356 136 | 4 | 156 1346 36 | 146 356 2 | 3 +----------------+----------------+ | 34 2 5 | 16 136 1346 | 2 | 346 36 1 | 5 2 34 | 1 +----------------+----------------+ a b c d e f Sudoku-X
there is an ALS-xz move that eliminates (3)a6, (3)f2.
This gives one placement and other eliminations from basic moves.
Step 10: in the following configuration
+-----------------+----------------+ | 16 13 36 | 2 4 5 | 6 | 2 5 4 | 3 16 16 | 5 +-----------------+----------------+ | 156 134 2 | 146 356 316 | 4 | 156 134 36 | 146 35 2 | 3 +-----------------+----------------+ | 34 2 5 | 16 136 146 | 2 | 34 6 1 | 5 2 34 | 1 +-----------------+----------------+ a b c d e f Sudoku-X
there is a fish on 3 eliminates (3)e3, (3)b5, resulting
in a configuration solvable with only singles (ste).
It would be nice to find a shorter path.
-------------------------
Update 2 (Feb 5, 2025): one way to find move 8 is to realize that
+---------------------+-------------------+ | 136 136 36 | 2 4 5 | 6 | 25 25 4 | 3 16 16 | 5 +---------------------+-------------------+ | 1356 1346 (25) | 146 356 1236 | 4 | 1356 @1346 (2)36 | 146 356 26 | 3 +---------------------+-------------------+ | 346-2 ((2)3) (56) | 156 #(36)1 1346 | 2 | 346 356 1 | 56 2 34 | 1 +---------------------+-------------------+ a b c d e f Sudoku-X
the chain (2 = 3)b2 - (3 =' 6)e2 - (6 = 5)c2 - (5 = 2)c4 - c3 = (2)b2
almost works, but there is a spoiler (1)e2. The remaining work
is to show that (1)e2 also leads to the elimination of (2)a2.
One proceeds going up diagonally (\), realizing that (1)e2 eliminates
(1)a6, so (1)b6 would be true and that eliminates (1)b3. Where to go
from there ? If (1)b3 is false, then it remains to show that each
remaining digit at b3 being true leads to -2 a2. For the digit 3 this is
immediate, since (3)b3 implies (2)b2.
Almost chains are very useful in finding krakens.
Btw, a reference on AIC (Alternative Inference Chain) here.
The source for AIC is Myth Jellies Thread.
--------------------------------------------
Update 3 (Feb 16m 2025): I posted the complete solution path here.
I propose the following 6x6 friendly ("Three birds") Sudoku-X puzzle.
(generated and displayed with Sudoku6Explainer)
After basics moves, one should get the following configuration:
+-------------------+-------------------+
| 12456 1345 12346 | 46 3456 356 | 6
| 3456 45 346 | 1 3456 2 | 5
+-------------------+-------------------+
| 1345 1345 145 | 26 1256 156 | 4
| 125 6 125 | 3 15 4 | 3
+-------------------+-------------------+
| 13456 145 3456 | 246 1246 136 | 2
| 1346 2 346 | 5 1346 16 | 1
+-------------------+-------------------+
a b c d e f Sudoku-X
Update (Feb 9, 2025)
Step 1: (above) there exists a S-wing that gives two placements and basic eliminations.
+----------------+-------------------+
| 126 13 1236 | 4 356 35 | 6
| 346 5 346 | 1 36 2 | 5
+----------------+-------------------+
| 1345 134 14 | 26 125 156 | 4
| 125 6 125 | 3 15 4 | 3
+----------------+-------------------+
| 1356 14 356 | 26 124 136 | 2
| 134 2 36 | 5 134 16 | 1
+----------------+-------------------+
a b c d e f Sudoku-X
Step 2: there exists an ALS W-wing that gives 7 placements and basic eliminations.
+-------------+---------------+
| 26 13 12 | 4 36 5 | 6
| 4 5 36 | 1 36 2 | 5
+-------------+---------------+
| 35 13 4 | 26 125 16 | 4
| 25 6 12 | 3 15 4 | 3
+-------------+---------------+
| 16 4 5 | 26 12 3 | 2
| 13 2 36 | 5 4 16 | 1
+-------------+---------------+
a b c d e f Sudoku-X
Step 3: this puzzle is now solvable with a Y-wing.
I propose the following tough Sudoku-X puzzle (SE = 7.6) with just 5 givens.
(generated and displayed with Sudoku6Explainer)
After basics we get (no additional digit!)
+----------------------+----------------------+
| 2356 12356 1235 | 12456 1245 126 | 6
| 2456 125 1245 | 3 125 126 | 5
+----------------------+----------------------+
| 1 236 236 | 24 234 5 | 4
| 2345 2345 235 | 12 6 123 | 3
+----------------------+----------------------+
| 23456 123456 12456| 1256 1235 1236 | 2
| 2356 12356 1256 | 1256 1235 4 | 1
+----------------------+----------------------+
a b c d e f Sudoku-X
Update 1 (May 1, 2025)
This puzzle is solvable in 5 steps. In my first step,
I found a fish (1) that gives -1 b5 [and a locked candidates elimination],
i.e.,
+----------------------+----------------------+
| 2356 12356 1235 | 12456 1245 126 | 6
| 2456 25 1245 | 3 125 126 | 5
+----------------------+----------------------+
| 1 236 236 | 24 234 5 | 4
| 2345 2345 235 | 12 6 123 | 3
+----------------------+----------------------+
| 23456 123456 12456| 256 1235 1236 | 2
| 2356 12356 1256 | 256 1235 4 | 1
+----------------------+----------------------+
a b c d e f Sudoku-X
Now, there exists another fish that was available before.
Update 2 (May 2, 2025)
The second fish was caught on digit (6), eliminating (6)a6
[it gives 1 placement, that created locked candidates and -3 a2].
The resulting configuration is
+----------------------+----------------------+
| 235 12356 1235 | 12456 1245 126 | 6
| 2456 25 1245 | 3 125 126 | 5
+----------------------+----------------------+
| 1 23 6 | 24 234 5 | 4
| 2345 2345 235 | 12 6 123 | 3
+----------------------+----------------------+
| 2456 123456 1245 | 256 1235 1236 | 2
| 2356 12356 125 | 256 1235 4 | 1
+----------------------+----------------------+
a b c d e f Sudoku-X
In the next move, one needs to study carefully the
pattern of digit (3), where a couple eliminations
are available.
Update 3 (May 3, 2025)
After move 3 (a fish loop !), one gets: -3 a3, b2, b3, b6, e1
+-----------------------+----------------------+
| 235 1256-3 1235 | 12456 1245 126 | 6
| 2456 25 1245 | 3 125 126 | 5
+-----------------------+----------------------+
| 1 23 6 | 24 234 5 | 4
| 245-3 245-3 235 | 12 6 123 | 3
+-----------------------+----------------------+
| 2456 12456-3 1245 | 256 1235 1236 | 2
| 2356 12356 125 | 256 125-3 4 | 1
+-----------------------+----------------------+
a b c d e f Sudoku-X
The next move involves studying the pattern of digit 6.
It leads to: -6 d6 AND -6 a2 [and basics].
The resulting configuration is
+------------------+----------------+
| 235 1256 1235 | 245 1245 26 | 6
| 2456 25 1245 | 3 125 126 | 5
+------------------+----------------+
| 1 23 6 | 24 234 5 | 4
| 245 245 235 | 1 6 23 | 3
+------------------+----------------+
| 245 1245 1245 | 6 235 123 | 2
| 36 36 125 | 25 125 4 | 1
+------------------+----------------+
a b c d e f Sudoku-X
Now, there is a way to finish the puzzle,
but it no longer involves a single digit move.
Update (May 4, 2025)
The last step (move 4) was a powerful Kraken Fish Diagonal (on 6).
A neat final blow involves finding a Y-wing in the previous
configuration displayed.
Let me propose one more Sudoku-X puzzle (SE = 7.7).
It is a bit harder than the last one.
(generated and displayed with Sudoku6Explainer)
After basics we get
+-------------------+-------------------+
| 25 1356 4 | 156 12356 1356 | 6
| 135 256 356 | 1456 1235 13456 | 5
+-------------------+-------------------+
| 345 345 1 | 456 456 2 | 4
| 6 245 25 | 3 145 145 | 3
+-------------------+-------------------+
| 12345 13456 2356 | 156 456 1356 | 2
| 1345 13456 356 | 2 1356 456 | 1
+-------------------+-------------------+
a b c d e f Sudoku-X
Update 1 (May 6, 2025)
For move 1, (25)a6 and (25)c3 indicate the possibility of
a W-wing, which indeed exists and eliminates -5 a4, -5 c5,
AND -5 a1, -5 f6 (diagonal effect). We get the grid
+--------------------+--------------------+
| 25 1356 4 | 156 12356 136 | 6
| 135 256 36 | 1456 1235 13456 | 5
+--------------------+--------------------+
| 34 345 1 | 456 456 2 | 4
| 6 245 25 | 3 14 145 | 3
+--------------------+--------------------+
| 1235-4 1346 2356 | 156 456 1356 | 2
| 134 13456 356 | 2 1356 456 | 1
+--------------------+--------------------+
a b c d e f Sudoku-X
Move 2 is the tough move of this puzzle!
Can you find a way to justify: -4 a2 ?
Update 2 (May 8, 2025)
For last move 2, if (6)d4 then we have the locked set (134)a1.b2.f6,
otherwise, we have 4 or 5 at c4 that can be used in a chain
to eliminate (4)a4. That elimination gives also
[&(4)a1|a4 => -4 d4, +4 d5 (first placement!), -4 b1]
+--------------------+--------------------+
| 25 1356 4 | 156 12356 136 | 6
| 135 256 36 | 4 1235 1356 | 5
+--------------------+--------------------+
| 34 345 1 | 56 456 2 | 4
| 6 25-4 25 | 3 14 415 | 3
+--------------------+--------------------+
| 1235 1346 2356 | 156 456 1356 | 2
| 134 1356 356 | 2 1356 456 | 1
+--------------------+--------------------+
a b c d e f Sudoku-X
For move 3, one can search for a skyscraper.
Update 3 (May 9, 2025)
First, I added a small correction to the previous update
[locked set (134)a1.b2.f6 instead of ALS (134)a1.b2.f6]
For move 3, there is a skyscraper on digit 4 that eliminates
(4)b4, (4)e4, followed by basic moves (NP and LC eliminations).
We reach the following configuration
+---------------------+--------------------+
| 25 1356 4 | 156 12356 136 | 6
| 135 256 36 | 4 123 1356 | 5
+---------------------+--------------------+
| 34 34 1 | 56 56 2 | 4
| 6 25 25 | 3 14 14 | 3
+---------------------+--------------------+
| 1235 1346 2356 | 156 456 1356 | 2
| 134 1356 356 | 2 1356 456 | 1
+---------------------+--------------------+
a b c d e f Sudoku-X
Now, there is a simple fish elimination, followed
by a two-digit wing. After that, only one more move
remains to be found.
Final Update (May 11, 2025)
Moves 4 and 5 can be written as
4. Grouped S-wing (1)a1.b1 = (1)e1 - (1=4)e4 - (4)e2 = (4)b2 => -1 b2
5. Fish (1): a5 = b8 - d6 = d2 => -1 a2 [& (1)a1|b1 => -1 e1; (1)a1|a5 => -1 e5]
Now,
+-------------------+--------------------+
| 25 1356 4 | 56-1 2356-1 136 | 6
| 135 256 36 | 4 23 1356 | 5
+-------------------+--------------------+
| 34 34 1 | 56 56 2 | 4
| 6 25 25 | 3 14 4-1 | 3
+-------------------+--------------------+
| 235 346 2356 | 156 456 356-1 | 2
| 134 1356 356 | 2 356 456 | 1
+-------------------+--------------------+
a b c d e f Sudoku-X
for the final step, there is an easy fish loop,
again exploiting the diagonal a1-f6 to eliminate
(1)d6, (1)e6, (1)f3, (1)f2. The remaining is
easy (many singles, a locked candidate, and
then only singles to the end).
Yesterday, I solved a nice, pleasant Sudoku-X puzzle.
It is perfect for a sunday morning, with many hidden wings.
It can be solved (manually) in a few steps.
(generated and displayed with Sudoku6Explainer)
+--------------------+-------------------+
| 1236 2356 156 | 4 56 26 | 6
| 246 256 456 | 136 126 356 | 5
+--------------------+-------------------+
| 246 246 46 | 5 3 1 | 4
| 5 1 3 | 26 246 246 | 3
+--------------------+-------------------+
| 36 46 156 | 1236 1456 23456 | 2
| 146 356 2 | 136 456 3456 | 1
+--------------------+-------------------+
a b c d e f
Update (Sept 15, 2025)
My first step used a fish on digit 3 to eliminate (3)d1.
+--------------------+-------------------+
| 1236 2356 156 | 4 56 26 | 6
| 246 256 456 | 136 126 356 | 5
+--------------------+-------------------+
| 246 246 46 | 5 3 1 | 4
| 5 1 3 | 26 246 246 | 3
+--------------------+-------------------+
| 36 46 156 | 1236 1456 23456 | 2
| 146 356 2 | 16-3 456 3456 | 1
+--------------------+-------------------+
a b c d e f
This move "wakes up" a sleepy wing!
Update (Sept 16, 2025)
+--------------------+-------------------+
| 1236 2356 156 | 4 56 26 | 6
| 246 256 456 | 136 126 356 | 5
+--------------------+-------------------+
| 246 246 46 | 5 3 1 | 4
| 5 1 3 | 26 246 246 | 3
+--------------------+-------------------+
| 36 46 156 | 1236 1456 23456 | 2
| 146 356 2 | 16 456 3456 | 1
+--------------------+-------------------+
a b c d e f
Actually, here two wings are very useful: a W-wing and a XYZ-wing.
After applying both, one needs only to find a third wing (waken by the W-wing!)
to solve this puzzle.
Update (Sept 18, 2025)
+-------------------+-------------------+
| 1236 2356 156 | 4 56 26 | 6
| 246 256 456 | 13 126 356 | 5
+-------------------+-------------------+
| 246 246 46 | 5 3 1 | 4
| 5 1 3 | 26 246 24 | 3
+-------------------+-------------------+
| 36 46 156 | 1236 145 23456 | 2
| 146 35 2 | 16 456 35-4 | 1
+-------------------+-------------------+
a b c d e f
The final wing is a Y-wing that eliminates (4)f1,
creating the NP(35) b1.f1 that eliminates (5)e1, b5,
creating a NP(26) b5.d3 that unlocks the puzzle with
singles to the end.