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Welcome!
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 12 steps using well-known patterns.
The start (4 steps) uses well-known patterns. Other two patterns are proposed as Problems.
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 (some are shown in Problems below) that can be
used in other puzzles (and hopefully has a short and elegant proof).
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)
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 |
| . x . . . x .___. x . . |
| 2 1 x 2 x 2 |
| .___. . . .___. . . |
| 3 1 2 2 | |
| .___. x . . . . . . |
| 3 3 2 | 3 |
| .___. . . . . x .___. |
| 2 2 2 x 2 |
| . . . . . . . . |
| 2 2 2 2 |
| . . . x . . . . . |
| 3 | 1 x 1 2 2 |
| . .___. x . x . . . . |
'-------------------------------'
In the next steps (not shown), I used twice the argument that the number
of lines in a row/column must be even.
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, in two key configurations, further analysis is needed to advance.
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 |
'--------------------------------------------------------------------'
Find first a solution in two steps. Then, look if they can be
combined in a single step.
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