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This is a place chosen to make available some ideas (on puzzles, patterns, etc)
that seemed unsuitable for posting in a Forum, but some people may find interesting.
For difficult puzzles (like the next one), possible solutions will not be shown (only hints).
For other puzzles, the solution wil be hidden so that a serious attempt can be made
without direct access to a solution.
(hopefully this will make the puzzles more interesting for newcomers).
The following Classic Sudoku Puzzle (SER = 8.4) was generated in April, 2021
using HoDoKu for a training practice with als/ahs.
I solved it (manually) in two main steps
(i.e., apart from moves with scope in a single house - the "basics").
(image created using SudokuExplainer)
After performing the basics, find non-basic moves that solve this puzzle.
.--------------------------------------------------------------.
| 126 1269 5 | 4 79 3 | 8 179 1279 |
| 123 8 234 | 159 579 6 | 12479 134579 123579 |
| 7 1349 34 | 1589 2 58 | 6 13459 1359 |
|---------------------+----------------+-----------------------|
| 28 24 1 | 59 48 7 | 3 6 59 |
| 68 5 678 | 3 89 1 | 79 2 4 |
| 9 347 347 | 6 45 2 | 17 157 8 |
|---------------------+----------------+-----------------------|
| 13568 1367 9 | 2 36 458 | 147 13478 137 |
| 2358 237 2378 | 578 1 458 | 2479 34789 6 |
| 4 12367 23678 | 78 36 9 | 5 1378 1237 |
'--------------------------------------------------------------'
and (a) look at the pattern of 2s to get some eliminations,
(b) Now, study carefully boxes 1 (top-left) and 4 (middle-left).
A very strong Sudoku player found a great solution in one step.
(items (a),(b) are for a solution in two steps)
The following Jigsaw Sudoku Puzzle was generated in Feb, 2024
using 1to9only new release of SudokuExplainer.
The puzzle is rated SER = 8.0, and it can be solved (manually) in
four main steps.
(image created using SudokuExplainer)
The following 5x5 Latin Squares Puzzle (generated with SudokuExplainer)
is rated SER = 8.3, and can be solved (manually) in four main steps.
(image created using SudokuExplainer)
(columns counted from left to right and rows from top to bottom)
First write all candidates in each cell.
Step 1: We can eliminate 3 from r2c4 using a sequence of bivalued cells.
The configuration for this is:
,-------------------------,
| 23 13 5 123 4 |
| 1 345 24 235 23 |
| 34 2 14 13 5 |
| 245 145 3 125 12 |
| 235 135 12 4 123 | Latin squares
'-------------------------'
Step 2: the harder part! (the reason for that SE rating).
,---------------------,
| 23 13 5 123 4 |
| 1 345 24 25 23 |
| 34 2 14 13 5 |
| 245 145 3 125 12 |
| 235 135 12 4 123 | Latin squares
'---------------------'
The suggestion is to study carefully the grid (above)
and try to find a way to show that 2 is false at that same cell r2c4.
If Step 2 is done, the puzzle can be solved in two more moves.
For step 3, the configuration is:
,-------------------------,
| 23 13 5 123 4 |
| 1 34 24 5 23 |
| 34 2 14 13 5 |
| 45 45 3 12 12 |
| 235 135 12 4 123 | Latin squares
'-------------------------'
For the last move, there is a pretty, simple idea! (see configuration below)
,---------------------,
| 2 13 5 13 4 |5
| 1 34 24 5 23 |4
| 34 2 14 13 5 |3
| 45 45 3 2 1 |2
| 35 135 12 4 23 |1 Latin squares
'---------------------'
a b c d e
Write a detailed solution to the following interesting 6x6 Slitherlink puzzle:
(From Simon Tatham's Portable Puzzle Collection)
This puzzle (rated hard) can be solved (manually) in 7 steps using well-known patterns.
My first step (basics) ends here:
. x . . . .___. .
x 3 2 2
.___. . . . . .
| 3 2 |
. . . . . . .
3 | 2 1 2
. .___. x . . . .
2 x 0 x
. .___. x .___. . .
3 | | 3 3
. . . . . . .
2 2
. . . . . . .
Step 2:
. x . . . .___. .
x => | 3 2 2
.___. . . . . .
| 3 2 |
. . . . . . .
3 | 2 1 2
. .___. x . . . .
2 x 0 x
. .___. x .___. . .
3 | | 3 3
. . . . . . .
2 2
. . . . . . .
The justification for step 2 is:
. x .===.===. .___. .
x ] 3 ] 2 2
.___. x .===. . . .
| 3 | 2 |
.===. x . . . . .
x ] 3 | 2 1 2
? x .___. x . . . .
] 2 x x 0 x
.===.___. x .___. .___.
3 | | 3 3
. . . . . . .
2 2
. . . . . . .
and line ending at (?) has no place to go.
The idea of writing out a detailed solution path is to break the solution into n parts
where each part is a well-identified pattern that can be used in other puzzles
(each hopefully having a short and elegant proof).
An example of this is the pattern in the lower-right corner.
. . .___. . . | . . .___. .___. |
| 3 3 | | 3 | 3 |
. . . . . . | . . . . . . |
2 | => 2 x | |
. . . . . . | . . . .___.___. |
----------------------' ----------------------'
Proof: either possible configuration for the 3-cell in
the left gets into the left upper corner of 2, and so
it must leave 2 at the lower right corner, implying that
horizontal line at the corner, that can only go up.
Now that last line arrived at the lower right corner
of the 3-cell on the right, and so the two edges that
do not contain that corner must be true.
The "x" avoids a small 2 cells-loop contaning the "3"
and additional horizontal line at the bottom follows from this.
The final configuration is the following
. .___. .___.___.___.
| | 3 | 2 2 |
.___. .___. .___. .
| 3 2 | | |
.___.___. .___. .___.
3 | 2 | 1 2
.___.___. .___.___.___.
| 2 0 |
. .___. .___. .___.
| | 3 | | 3 | | 3
. . . . . .___.
| | 2 | | | 2 |
.___. .___. .___.___.
Write a detailed solution to the following hard 5x5 Hidoku puzzle:
(Found recently by Albert.Lang)
I found an interesting reasonably simple and short (manual) solution to this puzzle
and for this reason I decided to propose it here.
(the restrictive rule of including the first and last digits in the givens is waived)
This pattern has the smallest number of clues for a 5x5 Hidoku with unique solution.
.-----------------------. |. . . 6 . |5 | | |. . *7 . 13 |4 | | |. *8 x . . |3 | | |9 . . . . |2 | | |. . . . . |1 '-----------------------' a b c d e
After the forced start, the possible values for x are very limited. For instance, for x=11, we must have
a1=25,b1=24 and the connection of 13 to 24 demands 10 cells, but placing 12 will destroy any possible way.
After basics, find a move that solves this Classic Sudoku puzzle ("three waves", SER = 7.2):
(image created using SudokuExplainer)
This puzzle was found with Sudoku Architect when I was playing around with waves.
.-------------------------------------------------------.
| 3 19 5 | 169 4 7 | 2 169 8 |
| 2679 4 169 | 1369 8 2369 | 1379 5 167 |
| 2679 129 8 | 5 169 2369 | 1379 13679 4 |
|------------------+-----------------+------------------|
| 48 6 34 | 7 5 1 | 38 2 9 |
| 1 79 2 | 689 3 689 | 4 67 5 |
| 5 3789 39 | 2 69 4 | 1378 1367 167 |
|------------------+-----------------+------------------|
| 49 5 1349 | 139 7 39 | 6 8 2 |
| 2689 1289 7 | 1689 169 5 | 19 4 3 |
| 689 1389 1369 | 4 2 3689 | 5 179 17 |
'-------------------------------------------------------'
Look for a fish in the digit 6 !After basics, this Classic Sudoku puzzle (SER = 8.3) can be solved in just two non-basic steps:
(image created using SudokuExplainer)
This puzzle was found with Sudoku Architect.
.------------------------------------------------------------.
| 1 4 59 | 67 2 8 | 56 79 3 |
| 2735 6 35 | 4 9 17 | 15 8 27 |
| 279 79 8 | 5 3 16 | 16 4 279 |
|--------------------+-------------------+-------------------|
| 356 8 1356 | 1367 1567 4 | 9 2 17 |
| 569 159 7 | 8 156 2 | 3 15 4 |
| 35 2 4 | 1379 157 579 | 8 157 6 |
|--------------------+-------------------+-------------------|
| 4 1379 169 | 1679 8 679 | 2 1369 5 |
| 5679 1579 1569 | 2 1567 3 | 4 169 8 |
| 8 1359 2 | 169 4 569 | 7 1369 19 |
'------------------------------------------------------------'
1) a nice UR (unique rectangle), 2) a symmetrical chain with 4 strong links.After some hard work, I solved (manually) the following 7x7 tough Slitherlink puzzle.
I hope that, by proposing it here, someone will have as much fun as I had solving it.
(From Simon Tatham's Portable Puzzle Collection)
x
1 3
_ _
3 1
x
helps to develop the upper-right part of this puzzle
(see Para's puzzle site)
.-------------------------------.
| . . . x . x .___.___.___. |
| 1 x | 2 x x | |
| . . . .___. x .___.___. |
| 1 x 1 | 3 x 1 x |
| . . . .___. x .___. x . |
| 3 | 1 x 2 | x |
| . . . . . x . x . . | |
| 3 2 x 2 | 2
| . . . . .___. x . . | |
| 1 x 1 3 | 2 3 | |
| . . x . x . . . . . |
| x 1 | 3 2 x |
| . . . x .___. . . . |
| | 3 1 x 2 |
| .___. . . . . . . |
'-------------------------------'
Now after analyzing the right side of the puzzle,
two simple inferences lead to substantial progress:
.-------------------------------.
| . . . x . x .___.___.___. |
| 1 x | 2 x x | | 7
| . . . .___. x .___.___. |
| 1 x 1 | 3 x 1 x | 6
| . . . .___. x .___. x . |
| x x 3 | 1 x 2 | x | 5
| . x .___. x .___. x . x . x . |
| | 3 | 2 x x | 2 x | 4
| . . .___. x .___. x .___. |
| 1 x 1 x | 3 | 2 x 3 | | 3
| . . x . x .___. x .___.___. |
| | x 1 | 3 x x 2 x x | 2
| .___. . x .___.___.___.___. |
| | 3 x x 1 x x 2 x | | 1
| .___.___.___.___.___.___.___. |
'-------------------------------'
a b c d e f g
Now looking at (1)c6, one finds the final inference
to finish the puzzle.
Final configuration:
.-------------------------------.
| .___.___. . .___.___.___. |
| | | 1 | 2 | |
| . .___. .___. .___.___. |
| | | 1 | 1 | 3 1 |
| . .___. .___. .___. . |
| | | 3 | 1 2 | |
| . .___. .___. . . . |
| | | 3 | 2 | 2 |
| . .___.___. .___. .___. |
| | 1 1 | 3 | 2 3 | |
| .___. . .___. .___.___. |
| | 1 | 3 2 |
| .___. . .___.___.___.___. |
| | 3 1 2 | |
| .___.___.___.___.___.___.___. |
'-------------------------------'
The following tough 7x7 Slitherlink puzzle has two blank rows and almost no standard patterns.
It was fun to find a way to solve it in a few steps.
(Interesting puzzle from the site www.themissingdocs.net)
.-------------------------------.
| . x . x . x . . . . x . |
| x 0 x x 1 X x |
| . x . x . X . . . . x . |
| x x 0 x 1 2 x 0 x |
| . . x . . . . x . x . |
| x x |
| . X . . . . .___.___. |
| 1 x 1 x 1 x | |
| . . . x .___. x . x . x . |
| 2 3 | 3 x 0 x 1 | |
| . . . . .___. x . x . |
| x | x | |
| . . . . x . x .___. x . |
| | 3 2 x 0 x 2 | | |
| .___. . . x . x . x .___. |
'-------------------------------'
The next two steps (not shown) allow a quick finish.
The following tough 7x7 Slitherlink is my first puzzle.
It was a lot of fun to create it full of patterns.
.-------------------------------.
| . . . .___. x .___. x . |
| | 3 | | 3 | 3 | x | 7
| . x . . . x .___. x . . |
| 2 1 x 2 x 2 | 6
| .___. . . .___. . . |
| 3 1 2 2 | | 5
| .___. x . . . . . . |
| 3 3 2 | 3 | 4
| .___. . . . . x .___. |
| 2 2 2 x 2 | 3
| . . . . . . . . |
| 2 2 2 2 | 2
| . . . x . . . . . |
| 3 | 1 x 1 2 2 | 1
| . .___. x . x . . . . |
'-------------------------------'
a b c d e f g
In the next 3 steps (not shown, but described), it is used twice the fact
that the number of lines in row 6 must be even.
Steps:
1) for the reason above, no line to the left of (2)g6;
2) a vertical line must exist at (2)a6 and, for the reason above,
no vertical line is true at (1)c6;
(3) next, the horizontal line at the top of (1)c5 leads to a quick contradiction.
After 3), it is easy to finish the puzzle.
Final configuration:
.-------------------------------.
| . .___. .___. .___. . |
| | 3 | | | 3 | 3 | |
| . . .___. .___. .___. |
| 2 | 1 2 2 | |
| .___. . .___.___. . . |
| | 3 1 | 2 | 2 | |
| .___. . .___. . .___. |
| 3 | 3 | 2 | | 3 |
| .___. . .___. . .___. |
| | 2 | 2 2 | 2 | |
| . . .___. .___. . . |
| | 2 | 2 | 2 2 | |
| .___. . . . . .___. |
| | 3 | 1 1 | 2 2 | |
| . .___. . .___.___. . |
'-------------------------------' JCO#1
I did not try to solve the puzzle without using that fact.
The following puzzle is my first 9x9 Hashi puzzle.
,-----------------------------------,
| 2 4 1 | 9
| |
| 2 1 2 2 | 8
| |
| 3 3 | 7
| |
| | 6
| |
| 2 4 3 | 5
| |
| 4 4 4 2 | 4
| |
| | 3
| |
| 1 2 3 2 | 2
| |
| 2 3 | 1 JCO#1 (9x9 Hashi)
'-----------------------------------'
a b c d e f g h i
(3)i1,(3)b7,(4)a4 are good places to start.
The sequence of inferences follow without obstruction,
leading quickly to
,-----------------------------------,
| *2*=====*4*---------*1* | 9
| | |
|*2*---------*1* | *2*---------*2*| 8
| | | | | |
| | *3*=========*3* | | | 7
| | | | | |
| | | | | | 6
| | | | | |
| | 2*-------------*4*=====*3* | | 5
| | | | |
|*4*=====*4*-----*4*=====*2* | | | 4
| | | | | | |
| | | | | | | 3
| | | | | | |
|*1* *2*-----*3*---------*2* | | 2
| | |
| *2*=========================*3*| 1
'-----------------------------------'
a b c d e f g h i
The following Classic Sudoku puzzle (SE = 7.3; found with Sudoku Architect)
is a tough puzzle! With careful analysis, it can be solved in a few steps,
revealing interesting structures forming its foundations.
There is a solution in 3 steps. The first step is complex and only boxes 1,7 do not take part.
,--------------------------------------------------------,
| 8 4 1279 | 6 17 157 | 239 125 2359|
| 127 5 127 | 3 9 178 | 4 128 6 |
| 3 169 169 | 158 2 4 | 89 7 59 |
|--------------------+-----------------+-----------------|
| 1279 12679 4 | 179 3 1679 | 5 26 8 |
| 1279 38 12679 | 1789 5 16789 | 2367 4 23 |
| 57 38 567 | 2 4 678 | 367 9 1 |
|--------------------+-----------------+-----------------|
| 4 1279 3 | 1579 8 (2)1579| 269 (25)6 (25)9|
| 259 29 8 | 4 6 (2)59 | 1 3 7 |
| 6 1279 12579 | 1579 17 3 | 289 258 4 |
'--------------------------------------------------------'
The key idea is to notice the strong link (2)r7c6 = (2)r8c6 and the almost naked pair(25)r7c89,
with spoilers (6)r7c8, (9)r7c9. After the resulting key elimination we get 4 placements.
Step 2 is easier and the key idea starts with (5)r9c5=(5)r9c8.
,------------------------------------------------------,
| 8 4 1279 | 6 17 157 | 239 125 2359|
| 12 5 127 | 3 9 178 | 4 128 6 |
| 3 169 169 | 158 2 4 | 89 7 59 |
|-----------------+------------------+-----------------|
| 129 126 4 | 179 3 1679 | 5 26 8 |
| 129 38 126 | 189 5 1689 | 7 4 23 |
| 7 38 5 | 2 4 68 | 36 9 1 |
|-----------------+------------------+-----------------|
| 4 1279 3 | 1579 8 12579 | 269 256 259 |
| 5 29 8 | 4 6 29 | 1 3 7 |
| 6 1279 129 | (5)179 17 3 | 289 (5)28 4 |
'------------------------------------------------------'
Boxes 1,4,5 and 7 are not used in this second step and the resulting elimination gives
7 placements, weakening severely the puzzle. A chain with 4 strong links finishes it.
Can you find it ?
,---------------------------------------------,
| 8 4 129 | 6 7 5 | 239 12 239 |
| 12 5 7 | 3 9 18 | 4 128 6 |
| 3 69 169 | 18 2 4 | 89 7 5 |
|--------------+--------------+---------------|
| 19 26 4 | 179 3 179 | 5 26 8 |
| 129 38 126 | 189 5 1689 | 7 4 23 |
| 7 38 5 | 2 4 68 | 36 9 1 |
|--------------+--------------+---------------|
| 4 1 3 | 579 8 279 | 269 256 29 |
| 5 29 8 | 4 6 29 | 1 3 7 |
| 6 7 29 | 59 1 3 | 289 258 4 |
'---------------------------------------------'
The following Classic Sudoku puzzle (SE = 7.1; found with Sudoku Architect)
is an interesting puzzle, with many patterns. How to solve it ?
After basics, the configuration is as follows.
.---------------------------------------------------.
| 6 249 1249 | 49 89 5 | 7 138 38 |
| 47 8 479 | 479 3 1 | 6 5 2 |
| 157 57 3 | 6 78 2 | 4 18 9 |
|---------------------+------------+----------------|
| 57 1 6 | 2 79 3 | 589 4 578 |
| 8 23579 279 | 579 6 4 | 2359 379 1 |
| 23457 234579 2479 | 579 1 8 | 2359 6 357 |
|---------------------+------------+----------------|
| 9 6 8 | 3 5 7 | 1 2 4 |
| 347 347 5 | 1 2 9 | 38 378 6 |
| 1237 237 127 | 8 4 6 | 359 379 357 |
'---------------------------------------------------'
I found a solution in 3 steps.
Step 1 is an ALS-xz move that eliminates (8)r1c9 [3 placements].
Step 2 is a wing that was not available before step 1.
The final step is a bit complex (wing-related, very pretty!)
and becomes possible by the key elimination made in step 2.
The following Classic Sudoku puzzle (SE = 7.3; found with Sudoku Architect)
has a slightly higher rating than #13, but in my view it is a lot easier to solve.
How ?
After basics, the configuration is as follows.
.------------------------------------------------------------.
| 1 4 6 | 5 3 2 | 8 9 7 |
| 9 2 5 | 7 8 14 | 6 134 13 |
| 3 8 7 | 46 9 146 | 124 5 12 |
|---------------+----------------------+---------------------|
| 4 19 8 | 3 7 5 | 129 6 129 |
| 7 3 12 | 2468 246 9 | 145 148 158 |
| 6 5 29 | 1 24 48 | 7 348 389 |
|---------------+----------------------+---------------------|
| 58 6 19 | 289 125 3 | 159 7 4 |
| 58 7 3 | 4689 1456 468 | 159 2 15689 |
| 2 19 4 | 689 156 7 | 3 18 15689 |
'------------------------------------------------------------'
I have found a solution in two steps. In the first step, I used an ALS
in column 6 within a chain (with 6 strong links) to show that 5 is false at r9c9.
After this, there are at least two wings that finish the puzzle.
The following tough Classic Sudoku puzzle (SE = 8.3; found with Sudoku Architect)
has a solution in two steps that requires considerable work analyzing the puzzle.
After basics, the configuration is as follows.
.--------------------------------------------------------------------.
| 23567 2367 2369 | 469 479 1 | 267 25679 8 |
| 1267 8 4 | 69 5 679 | 3 12679 279 |
| 1567 167 69 | 2 3 8 | 1467 15679 4579 |
|----------------------+----------------------+----------------------|
| 2367 2367 236 | 359 29 4 | 8 3569 1 |
| 48 5 1 | 389 6 79 | 247 2379 23479 |
| 48 9 236 | 1 278 57 | 467 3567 3457 |
|----------------------+----------------------+----------------------|
| 1269 1246 7 | 45689 1489 569 | 12 1238 23 |
| 126 1246 5 | 468 148 3 | 9 1278 27 |
| 139 13 8 | 7 19 2 | 5 4 6 |
'--------------------------------------------------------------------'
For the first move, search for a Sue de Coq!
(SDCs are elusive, so look carefully at boxes 5 and 6)
The second move is harder: only two boxes are not involved.
(ALS(193)r9c25 and ALS(2379)r5c678 take part in this powerful move)
The following Classic Sudoku puzzle (SE = 7.1; found with Sudoku Architect)
has a nice solution in one step.
After basics, the configuration is as follows.
.-----------------------------------------------------------.
| 1 238 238 | 6 7 9 | 4 23 5 |
| 45 29 7 | 45 3 8 | 6 29 1 |
| 459 369 3456 | 2 1 45 | 8 39 7 |
|-------------------+-------------------+-------------------|
| 7 4 9 |(35) 8 6 | 235 1 23 |
| 8 123 123 | 1359 (59) 7 | 359 6 4 |
| 6 5 13 | 1349 2 34 | 39 7 8 |
|-------------------+-------------------+-------------------|
| 2 689 68 | 379 4 1 | 37 5 369 |
| 459 19 145 | 3579 6 235 | 1237 8 239 |
| 3 7 156 | 8 59 25 | 12 4 269 |
'-----------------------------------------------------------'
ALS(359) (shown above) participates in a powerful move that essentially solves the puzzle.
I have battled with a nice Slitherlink puzzle (hard) from Simon Tatham's collection.
I have found a solution in 4 main steps: one can make good progress at the start,
using the two 3s at the corners, the sole "0" and that one cannot have
a line between 1 and 2 in column "g", etc. This start leads to
a long sequence of forced moves.
.-------------------------------.
| . . . . . .___.___. |
| 2 | 3 3 | |
| . . . . x .___. .___. |
| 1 2 1 x x |
| . . . . . . .___. |
| 2 2 1 3 | |
| . . . . . . x .___. |
| 2 | x |
| . . x . . . . x .___. |
| 3 | 1 x 1 1 x 0 x 2 | |
| .___. x . . . x . x . x . |
| x 1 | 1 x x 1 | |
| . . . .___. x .___. x . |
| | 3 2 2 x 2 | 3 | | |
| .___. . .___.___. .___. |
'-------------------------------'
a b c d e f g
The position above is still part 1 (with basic moves still available).
Later on, further analysis is needed to advance (see next).
.-------------------------------.
| . . . x .___. .___.___. |
| 2 | | 3 | 3 | | 7
| . . . . x .___. x .___. |
| 1 2 1 x x | x | 6
| . . . . . x . x .___. |
| | 2 2 1 x 3 | | 5
| . x . . . . . x .___. |
| | | 2 | x | 4
| . . x . . . . x .___. |
| | 3 | 1 x 1 1 x 0 x 2 | | 3
| .___. x . x . . x . x . x . |
| x x 1 x x | 1 x x 1 | | 2
| .___.___.___.___. x .___. x . |
| | 3 2 2 x 2 | 3 | | | 1
| .___.___.___.___.___. .___. |
'-------------------------------'
a b c d d f g
In the configuration above, (1r)c3 leads to a contradiction.
[(1r)c3 means a line to the right of the cell c3 (it has a 1)]
[The contradiction was: three lines arriving at (2)b5]
Now
.-------------------------------.
| . . . x .___. .___.___. |
| 2 | | 3 | 3 | | 7
| . . .___. x .___. x .___. |
| 1 2 x 1 x x | x | 6
| . . . .___. x . x .___. |
| | 2 2 | 1 x 3 | | 5
| . x . . . . x . x .___. |
| | | | 2 x | x | 4 already four vertical lines at line 4,
| . . x .___. . x . x .___. | so _
| | 3 | 1 x 1 x | 1 x 0 x 2 | | 3 |2
| .___. x . x . x . x . x . x . | and
| x x 1 x x | 1 x x 1 | | 2 2 |
| .___.___.___.___. x .___. x . | ___.
| | 3 2 2 x 2 | 3 | | | 1
| .___.___.___.___.___. .___. | are not possible for (2)d4 !
'-------------------------------'
a b c d e f g
Next, one can show that (2l)d4 and (2r)d4 true lead quickly to a contradiction.
So, (2b)d4 and (2t)d4 must be true [notation: l=left,r=right,t=top,b=bottom]
Now the puzzle is finished easily using once more that idea
of having an even number of lines in a row or column.
Final configuration:
.-------------------------------.
| .___.___. .___. .___.___. |
| | 2 | | | 3 | 3 | |
| . . .___. .___. .___. |
| | 1 2 1 | |
| . .___.___.___. . .___. |
| | 2 | 2 | 1 3 | |
| . . .___.___. . .___. |
| | | | 2 | |
| . . .___.___. . .___. |
| | 3 | 1 1 | 1 0 2 | |
| .___. . . . . . . |
| 1 | 1 1 | |
| .___.___.___.___. .___. . |
| | 3 2 2 2 | 3 | | |
| .___.___.___.___.___. .___. |
'-------------------------------'
The following Classic Sudoku puzzle (SE = 7.3; found with Sudoku Architect)
has an interesting solution in two steps.
After basics, the configuration is as follows.
,-----------------------------------------------------------,
| 19 69 8 | 2 5 7 | 4 3 16 |
| 25 27 4 | 6 3 1 | 9 8 57 |
| 15 67 3 | 4 8 9 | 15 2 67 |
|-------------------+-------------------+-------------------|
| 247 8 257 | 1357 6 245 | 15 9 135 |
| 479 3 579 | 1579 179 458 | 2 6 158 |
| 6 29 1 | 35 29 58 | 7 4 358 |
|-------------------+-------------------+-------------------|
| 3 1 79 | 79 4 6 | 8 5 2 |
| 8 4 6 | 15 12 25 | 3 7 9 |
| 279 5 279 | 8 79 3 | 6 1 4 |
'-----------------------------------------------------------'
My first solution had 2 steps:
(i) UR(27)r49c13 using externals to eliminate 7 from r7c4 and from r5c5,
(ii) a simple chain with 5 strong links using (5)r3c7 = (5)r4c7
(eliminates 9 from r1c2)
Then, I noticed a solution in just one step, using
,-----------------------------------------------------------,
| 19 69 8 | 2 5 7 | 4 3 16 |
| 25 27 4 | 6 3 1 | 9 8 57 |
| 15 67 3 | 4 8 9 | 1(5) 2 67 |
|-------------------+-------------------+-------------------|
| 247 8 257 | 1357 6 245 | 1(5) 9 135 |
| 479 3 579 | 1579 179 458 | 2 6 158 |
| 6 2-9 1 | 35 29 58 | 7 4 358 |
|-------------------+-------------------+-------------------|
| 3 1 (7)9 |(7)9 4 6 | 8 5 2 |
| 8 4 6 | 15 12 25 | 3 7 9 |
| 279 5 279 | 8 79 3 | 6 1 4 |
'-----------------------------------------------------------'
the strong links (5)r3c7 = (5)r4c7 and (7)r7c3 = (7)r7c4.
I have recently battled with the following tough Classic Sudoku puzzle
(SER = 8.4; found with Sudoku Architect)
Perhaps someone else is also in the mood for such battle.
After basics, the configuration is as follows.
.--------------------------------------------------------------------.
| 9 7 234 | 5 13 8 | 124 146 1246 |
| 268 236 2368 | 4 7 13 | 128 5 9 |
| 1 45 458 | 9 6 2 | 3 47 478 |
|----------------------+----------------------+----------------------|
| 24 234 1 | 7 29 6 | 459 8 345 |
| 5 8 23467 | 1 29 34 | 479 467 3467 |
| 467 346 9 | 8 34 5 | 147 2 13467 |
|----------------------+----------------------+----------------------|
| 478 1 478 | 2 5 9 | 6 3 478 |
| 3 245 2457 | 6 8 14 | 12457 9 12457 |
| 268 9 2568 | 3 14 7 | 258 14 258 |
'--------------------------------------------------------------------'
After basics, x-wing(1), ALS-xz(column 8, box 3), Kite(4) and ER(3),
one gets the following key configuration:
.--------------------------------------------------------------------.
| 9 7 234 | 5 13 8 | 24 14 6 |
| 268 26 2368 | 4 7 13 | 128 5 9 |
| 1 45 458 | 9 6 2 | 3 7 48 |
|----------------------+----------------------+----------------------|
| 24 234 1 | 7 29 6 | 459 8 345 |
| 5 8 247 | 1 29 34 | 479 6 347 |
| 467 346 9 | 8 34 5 | 147 2 1347 |
|----------------------+----------------------+----------------------|
| 478 1 478 | 2 5 9 | 6 3 478 |
| 3 245 2457 | 6 8 14 | 1457 9 12457 |
| 268 9 2568 | 3 14 7 | 58 14 258 |
'--------------------------------------------------------------------'
Now there is something interesting (column 1 & box 4).
After the elimination from that step, there is an
x-chain on digit 4 that leads to
,--------------------------------------------------------------------,
| 9 7 23 | 5 13 8 | 24 14 6 |
| 268 26 2368 | 4 7 13 | 12 5 9 |
| 1 45 45 | 9 6 2 | 3 7 8 |
|----------------------+----------------------+----------------------|
| 24 23 1 | 7 29 6 | 459 8 345 |
| 5 8 247 | 1 29 34 | 479 6 347 |
| 467 36 9 | 8 34 5 | 147 2 1347 |
|----------------------+----------------------+----------------------|
| 478 1 478 | 2 5 9 | 6 3 47 |
| 3 45 2457 | 6 8 14 | 1457 9 12457 |
| 26 9 256 | 3 14 7 | 8 14 25 |
'--------------------------------------------------------------------'
(now rated SE = 7.1).
I have found an interesting tough Classic Sudoku puzzle
(SER = 8.3; found with Sudoku Architect)
This puzzle is solvable in two steps.
After basics, the configuration is as follows.
,--------------------------------------------------------------------,
| 267 2567 25 | 3 4 79 | 8 69 1 |
| 9 4 3 | 1 6 8 | 57 257 257 |
| 67 8 1 | 5 2 79 | 39 3469 46 |
|----------------------+----------------------+----------------------|
| 2478 27 248 | 267 9 3 | 4567 1 4567 |
| 5 2379 6 | 27 1 4 | 39 379 8 |
| 1 379 (49) | 67 8 5 | 2 34679 467 |
|----------------------+----------------------+----------------------|
| 468 56 7 | 48 3 2 | 1 456 9 |
| 248 1 24589 | 489 7 6 | 45 245 3 |
| 3 269 249 | 49 5 1 | 467 8 2467 |
'--------------------------------------------------------------------'
Hint for solution in two steps: for both steps, study the interaction
between (49)r6c3 and row 4.
This classic sudoku puzzle has SER = 8.4 but has a solution in just one step!
(found with Sudoku Architect)
After basics, the configuration is as follows.
,--------------------------------------------------------------------,
| 2 8 359 | 69 679 4 | 35 137 157 |
| 39 6 1 | 5 79 23 | 2348 23478 2478 |
| 35 7 4 | 13 8 123 | 235 6 9 |
|----------------------+----------------------+----------------------|
| 589 3 589 | 689 469 7 | 1 248 24568 |
| 4 2 5789 | 1689 369 16 | 3568 378 5678 |
| 6 1 78 | 2 34 5 | 9 3478 478 |
|----------------------+----------------------+----------------------|
| 138 4 2368 | 7 5 9 | 268 128 1268 |
| 18 9 (68) | 346 2 36 | 7 5 1468 |
| 7 5 26 | 46 1 8 | 246 9 3 |
'--------------------------------------------------------------------'
The idea is to find first a solution in two steps and then,
look if they can be combined in a single step.
My first move involves boxes 1,2,4,7,8; (68)r8c3 is a key part.
After that move,
,------------------------------------------------------,
| 2 8 35 | 69 69 4 | 35 17 17 |
| 9 6 1 | 5 7 23 | 2348 2348 248 |
| 35 7 4 | 1-3 8 123 | 235 6 9 |
|---------------+------------------+-------------------|
| 58 3 589 | 869 469 7 | 1 248 24568 |
| 4 2 5789| 1869 369 16 | 368 378 5678 |
| 6 1 78 | 2 34 5 | 9 3478 478 |
|---------------+------------------+-------------------|
| 138 4 2368| 7 5 9 | 268 128 1268 |
| 18 9 68 | 346 2 36 | 7 5 1468 |
| 7 5 26 | 46 1 8 | 246 9 3 |
'------------------------------------------------------'
there is an ALS-wing that solves the puzzle (boxes 1,2,4,5).
Regarding the solution in one main step, looking if move 2 is possible as the first
step, one notices a single spoiler. But the spoiler being true leads to the
same elimination of the ALS-wing of the previous 2-step solution.
(68)r8c3 (and box 7) does not take part in this second solution.
This classic sudoku puzzle has SER = 8.3 and has a solution in three steps!
(found with Sudoku Architect)
After basics, the configuration is as follows.
,-----------------------------------------------------------------------------,
| 9 8 6 | 5 247 237 | 1 37 347 |
| 457 3457 2 | 1 4679 367 | 8 35679 345679 |
| 1457 13457 1345 | 34679 4679 8 | 4579 35679 2 |
|-------------------------+-------------------------+-------------------------|
| 3 14 9 | 2467 5 267 | 247 1267 8 |
| 6 2 7 | 49 8 1 | 3 59 459 |
| 145 145 8 | 234679 24679 2367 | 2479 12679 14679 |
|-------------------------+-------------------------+-------------------------|
| 1247 9 134 | 267 1267 5 | 27 8 137 |
| 8 1357 135 | 27 127 4 | 6 123579 13579 |
| 1257 6 15 | 8 3 9 | 257 4 157 |
'-----------------------------------------------------------------------------'
For move 1, there is a beaultiful relashionship between column 7 and box 6.
This leads to four placements and a hidden triple.
For move 2,
,----------------------------------------------------------------------, | 9 8 6 | 5 247 237 | 1 37 34 | | 457 3457 2 | 1 4679 367 | 8 35679 34569 | | 1457 13457 435 | 34679 4679 8 | 479 35679 2 | |----------------------+-----------------------+-----------------------| | 3 14 9 | 2467 5 267 | 47 126 8 | | 6 2 7 | 49 8 1 | 3 59 459 | | 145 145 8 | 234679 24679 2367 | 479 126 16 | |----------------------+-----------------------+-----------------------| | 47 9 34 | 67 167 5 | 2 8 13 | | 8 357 35 | 27 127 4 | 6 139 139 | | 2 6 1 | 8 3 9 | 5 4 7 | '----------------------------------------------------------------------'
there is a nice wing, not available before those placements and eliminations.
For move 3,
,-----------------------------------------------------------,
| 9 8 6 | 5 427 27 | 1 37 34 |
| 457 3 2 | 1 9 67 | 8 567 456 |
| 4157 4157 45 | 3 467 8 | 9 567 2 |
|--------------------+------------------+-------------------|
| 3 14 9 | 267 5 267 | 47 126 8 |
| 6 2 7 | 4 8 1 | 3 59 59 |
| 145 145 8 | 9 267 3 | 47 126 16 |
|--------------------+------------------+-------------------|
| 47 9 34 | 67 167 5 | 2 8 13 |
| 8 57 35 | 27 127 4 | 6 139 139 |
| 2 6 1 | 8 3 9 | 5 4 7 |
'-----------------------------------------------------------'
there is an ALS-wing involving neighbour boxes.
Steps 1 and 2 can be replaced by a kraken-cell (almost M-wing).
This interesting Classic Sudoku (6x6) puzzle has SER = 6.6 and can be solved in three steps.
(puzzle found with Sukaku 6x6Explainer)
After basics, the configuration is as follows.
,-----------------------------------,
6 | 2 *16 3 | 156 15 4 |
5 |*16 4 5 | 2 3 *16 |
|----------------+------------------|
4 | 1456 2 146 | 1456 145 3 |
3 | 3 *(5)16 146 | 1456 1245 *(2)16 |
|----------------+------------------|
2 | 14 3 2 | 14 6 5 |
1 | 1456 156 146 | 3 124 12 |
'-----------------------------------'
a b c d e f
As usual, many ways are possible.
The strong link (5)b3=(2)f3 is nice
(internal guardians of the bivalued oddagon *(16))
and leads to placements: +5 b3, +5 a1 (how ?)
Move 2 is a simple skyscraper and move 3 is similar (same idea) to move 1.
This interesting Slitherlink (6x6) puzzle has a tough start.
(puzzle from Simon Tatham's Collection)
.---------------------------. | . . . . . . . | | 1 1 1 | | . . . . . . . | | 2 1 2 | | . . . . . . . | | 3 2 1 | | . . . . . . . | | 2 1 2 | | . . . . . . . | | 1 3 2 | | . . . . . . . | | 1 3 2 | | . . . . . . . | '---------------------------'
The following is a sketch of my solution in 3 steps:
Step 1:
.---------------------------. | . . . . . . . | | 1 x 1 x 1 | | . . . . . . . | | 2 1 2 | | . . . . . . . | | 3 2 1 | | . . . . . . . | | 2 x 1 2 | | . . . .___. x . . | | x x 1 3 | 2 | | . x . x . . . x . . | | x x 1 | 3 2 | | . x . x .___. . . . | '---------------------------'
Step 2 (4 parts):
to show that the line marked === below leads to contradiction. .---------------------------. | . . . . . . . | | 1 x 1 x 1 | | . . . . . . . | | 2 1 2 | | . . . . . . . | | 3 2 1 | | . . . . . . . | | 2 x 1 2 | | . .===. .___. x . . | | x x 1 x 3 | 2 | | . x . x . . . x . . | | x x 1 | 3 2 | | . x . x .___. . . . | '---------------------------'
Step 3:
to show that the line === shown below leads to contradiction. .---------------------------. | . . . . . . . | | 1 x 1 x 1 | | . . . x . . . . | | 2 1 x 2 | | . . . . . . . | | | 3 2 1 | | .___. . . . . . | | x x | 2 x 1 2 | | . x . x . x .___. x . . | | x x 1 | 3 | 2 | | . x . x . x .===. x . . | | x x 1 | 3 | 2 | | . x . x .___. x . . . | '---------------------------'
After this, the puzzle is solved quickly.
The final configuration is
.---------------------------. | .___.___.___.___.___.___. | | | 1 1 1 | | | .___.___. . . .___. | | 2 | 1 2 | | | .___.___. .___.___. . | | | 3 2 1 | | | .___.___. .___.___.___. | | | 2 1 2 | | | . . . .___. . . | | 1 | | 3 | 2 | | | . . . . . .___. | | 1 | 3 | 2 | | | | . . .___. .___. . | '---------------------------'
This classic sudoku puzzle has SE = 8.3
(it has a solution in 3 steps (and 2nd, 3rd are easy))!
(found with Sudoku Architect)
After basics
,-----------------------------------------------------------,
| 9 2 347 | 1 8 37 | 346 5 367 |
| 3678 468 3478 | 5 369 2 | 1 348 3789 |
| 3678 5 1 | 4 369 379 | 39 2 3789 |
|-------------------+-------------------+-------------------|
| 1 89 6 | 28 234 348 | 7 39 5 |
| 4 7 2 | 9 5 13 | 8 136 36 |
| 5 3 89 | 6 7 18 | 2 19 4 |
|-------------------+-------------------+-------------------|
| 36 469 349 | 28 249 489 | 5 7 1 |
| 2 489 5 | 7 1 6 | 349 348 389 |
| 78 1 4789 | 3 49 5 | 469 468 2 |
'-----------------------------------------------------------'
For move 1, look for interactions: blocks 2, 3 and 6.
In more detail: look for an almost Y-wing that almost solves the puzzle.
(needs something like two skyscrapers to finish)
The following (7x7 tough) Slitherlink puzzle is my second puzzle.
.-------------------------------.
| . . . . . . . . |
| 3 3 1 1 |
| . . . . . . . . |
| 1 2 |
| . . . . . . . . |
| 2 3 3 |
| . . . . . . . . |
| 2 |
| . . . . . . . . |
| 3 1 2 |
| . . . . . . . . |
| 3 3 1 |
| . . . . . . . . |
| 1 2 1 2 |
| . . . . . . . . | JCO#2
'-------------------------------'
Basics moves lead to
.-------------------------------.
| . . . .___. x . x . x . |
| 3 3 | 1 x x 1 x | 7
| . . . . . x .___.___. | (only way!)
| x 1 | x 2 | | 6
| . .___. x . . x .___. x . |
| 2 | 3 | 3 | | | 5
| . . x .___. x . .___. x . |
| x x 2 | | 4
| . .___. x . x . . x .___. |
| x 3 | x 1 | | 2 x | 3
| .___. . . . x . x . x . |
| | 3 x x x | 3 | | 1 x | 2
| .___.___. x . x .___. x . . |
| 1 x 2 | x 1 x x 2 | | 1
| . . x .___.___.___.___. . |
'-------------------------------' JCO#2
a b c d e f g
Following this, two more inferences were needed to get
.-------------------------------.
| .___.___. .___. . . . |
| | 3 | | 3 | 1 1 |
| . .___. . . .___.___. |
| | | 1 | | | 2 | |
| . .___. . . .___. . |
| | 2 | 3 | | 3 | | |
| .___. .___. . .___. . |
| | | | 2 | |
| . .___. . . . .___. |
| 3 | 1 | | | 2 |
| .___.___. . . . . . |
| | 3 | 3 | | 1 |
| .___.___. . .___. . . |
| 1 2 | 1 2 | |
| . . .___.___.___.___. . | JCO#2
'-------------------------------'
This classic sudoku puzzle has SE = 8.3 (and a solution in just 2 steps)!
(found with Sudoku Architect)
After basics
,------------------------------------------------------------------,
| 1 79 2 | 3 6 4 | 89 789 5 |
| 58 6 39 | 15789 2 1579 | 139 4 379 |
| 58 3479 349 | 15789 158 1579 | 6 1279 2379 |
|--------------------+------------------------+--------------------|
| 347 234 5 | 26789 348 23679 | 3489 289 1 |
| 347 8 134 | 12579 1345 123579 | 3459 6 2349 |
| 9 1234 6 | 1258 13458 1235 | 7 258 234 |
|--------------------+------------------------+--------------------|
| 34 1349 7 | 26 135 26 | 1459 159 8 |
| 2 5 19 | 4 7 8 | 19 3 6 |
| 6 134 8 | 15 9 135 | 2 157 47 |
'------------------------------------------------------------------'
(Jan 01, 2025)
Look carefully at Box 3 and Row 2.
(Why these two houses ? The three "39s" in Row 2 are a sort of clue.)
Happy New Year !!
I propose the following puzzle to be solved * without pencilmarks *
i.e., without writing candidates in the cells.
(found with Sudoku Architect)
Using the given digits, the following configuration should be obtained.
Digits marked with ** are derived digits (to distinguish from the given digits).
,---------+---------+---------,
| 6 3 8 | . 5 . | 4 *2* 9 |
| 7 1 9 |*2* 3 4 | . 5 . |
| . . *2*| . 8 . | 1 . . |
|---------+---------+---------|
| 2 9 . | . 6 . | 5 . 4 |
| . . 5 | 3 . *2*| 9 . . |
|*8* . . | . 9 . |*2* . *7*|
|---------+---------+---------|
| . . *4*| . . . |*7* 9 . |
| . 8 . | . . . | . *4* . |
| 9 . . | . . . | . . 1 |
'-----------------------------'
Update (Jan 6, 2025):
Step 1: a key digit can be placed as a consequence of
,---------+---------+---------,
| 6 3 8 | . 5 . | 4 *2* 9 | (17)r1c46
| 7 1 9 |*2* 3 4 | . 5 . | (68)r2c79
| . . *2*| . 8 . | 1 . . | (69)r3c46
|---------+---------+---------|
| 2 9 . | . 6 . | 5 . 4 |
| . x 5 | 3 x *2*| 9 . . | --.
|*8* . . | . 9 . |*2* . *7*| |
|---------+---------+---------| |----- 4 can only go here at column 5
| . . *4*| . . . |*7* 9 . | | 7 can only go here at column 2
| . 8 . | . . . | . *4* . | |
| 9 x . | . x . | . . 1 | --'
'-----------------------------'
^ ^
'---------'
2 can only go here at row 9
7 can only go here at row 5
Step 2: search for a unique rectangle.
I propose the following tough puzzle (SE = 8.3).
(found with Sudoku Architect)
Study carefully boxes 3,6,9 of the configuration after basics.
,--------------------------------------------------------------------------------------,
| 13469 1347 34679 | 358 568 2 | 13459 1389 1345689 |
| 2369 5 8 | 7 1 4 | 39 239 2369 |
| 12346 1234 2346 | 358 9 356 | 1345 1238 7 |
|----------------------------+----------------------------+----------------------------|
| 34568 3478 3467 | 1 45678 35679 | 2 379 349 |
| 123468 9 23467 | 348 4678 367 | 1347 5 134 |
| 1345 1347 347 | 3459 2 3579 | 8 6 1349 |
|----------------------------+----------------------------+----------------------------|
| 239 23 5 | 249 47 8 | 6 12379 1239 |
| 7 248 249 | 6 3 1 | 59 289 2589 |
| 2389 6 1 | 259 57 579 | 379 4 2389 |
'--------------------------------------------------------------------------------------'
update (Jan 18, 2025)
This difficult puzzle is solvable in 4 steps. The first step is the hardest to see.
One good start is the bivalued cell (59)r8c7 and the observation that at box 3,
r12c9 is an AHS (6 is locked). There is a powerful Loop with cells in boxes 1 and 3.
This was my way to find a Sue de Coq in this puzzle.
Another way (no loop) involves only boxes 6 and 9: find two ALS connected to an AALS.
It is important to pay attention to all possible eliminations: -9 r8c3 is crucial!
After that important move, the configuration is the following
,---------------------------------------------------------------------,
| 1346 7 9 | 358 568 2 | 1345 138 568 |
| 236 5 8 | 7 1 4 | 39 239 26 |
| 12346 1234 2346 | 358 9 356 | 1345 1238 7 |
|----------------------+-----------------------+----------------------|
| 34568 348 3467 | 1 45678 35679 | 2 37 349 |
| 123468 9 23467 | 348 4678 367 | 137 5 134 |
| 1345 134 347 | 3459 2 3579 | 8 6 1349 |
|----------------------+-----------------------+----------------------|
| 239 23 5 | 249 47 8 | 6 137 123 |
| 7 248 24 | 6 3 1 | 59 289 258 |
| 2389 6 1 | 259 57 579 | 37 4 238 |
'---------------------------------------------------------------------'
Looking at the eliminations from move 1, there is a wing at boxes 6,9 (two wings, actually).
For move 3, study carefully the configuration of 3.
The final move involves boxes 2 and 3.
I have been solving Sudoku-X puzzles lately.
The following 6x6 Sudoku-X puzzle has nice features (SE = 7.1).
(generated and displayed with Sudoku6Explainer)
After basics moves, we get the following configuration.
+-----------------------+--------------------+
| 12346 23456 13456 | 346 1235 456 | 6
| 13456 2346 13456 | 346 235 1456 | 5
+-----------------------+--------------------+
| 245 245 24 | 1 *6* *3* | 4
| 36 *1* 36 | 5 *4* 2 | 3
+-----------------------+--------------------+
| 13456 3456 13456 |*2* 13 1456 | 2
| 245 23456 123456 | 346 135 146 | 1
+-----------------------+--------------------+
a b c d e f Sudoku-X
(numbers with stars are givens)
In the game called Slitherlink, show all the details
proving that the following configuration is impossible
(My solution has a trivial initial observation, an easy Case 1 and Case 2 divided in two sub-cases)
We analyze the possible outcomes from this configuration but can already discard
all configurations in which the lower "2"-cell has vertical lines, or horizontal lines,
in both edges.
Also, this configuration
.------------------- | . . . . . | 2 | . . . . . | 2 | .___. . . . | | 2 |A. . . . . | x | . . . . .
and this
.------------------- | . . . . . | 2 | . . . . . | 2 | . . . . . | 2 | |A.___. . . . | x | . . . . .
are clearly impossible,
(dead end at corner A).
Case 1:
.-------------------
| . . . . .
| 2
| . . . . .
| 2
| .___. . . .
| 2 |
| . . . . .
| x
| . . . . .
Now the following is forced
.-------------------
| . .___.___. .
| x 2 |
| . .___. x . .
| | x 2 |
| .___. x . . .
| 2 |
| . . . . .
| x
| . . . . .
and this leads to an impossibility in the cell at the corner, reached by three lines.
.------------------- | . . . . . | 2 | . . . . . | 2 | . . . . . | | 2 | .___. . . . | x | . . . . .
The following is a forced move
.------------------- | . . . . . | 2 | . .B . . . | | 2 | . x .A . . . | | 2 x | .___. . . . | x | . . . . .
Case 2.1: If a vertical line exists at AB, it forces
.------------------- | . . . . . | | 2 | .C . . . . | | | 2 | . x .___.D . . | | 2 x | .___. . . . | x | . . . . .
and any continuation for the line meeting corner C will create a loop (impossible).
Case 2.2: the other possibility is to have the edge AB blocked, and then the bottom edge AD
of the "2"-cell in the middle needs to be blocked too, forcing the following configuration to be reached.
.------------------- | . .___.___. . | x 2 | | . .___. x . . | | x 2 | | . x . x . . . | | 2 x | .___. . . . | x | . . . . .
Now, we have an impossible situation in the cell at the corner, reached by three lines. □
The following Jigsaw Sudoku Layout is invalid. Why ?
(image created using SudokuExplainer)
A Jigsaw Sudoku Layout is invalid if no Latin Squares exist for that Layouts.
Please find a short, yet complete, explanation (no computations involved).
To my knowledge this question was first posed, and subsequently pursued, by Mathimagics.
In the configuration below (of a HEX game), it is blue's turn.
Blue has a decisive advantage. How to win ?
Contact: sudo.jco.br@gmail.com