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The Jigsaw Sudoku Layout below is invalid (no Latin Squares exist for it). Why?

Please find a short, yet complete, explanation (no computations involved).

+-------------------+---------------+
| .   .   .   .   . | .   .   .   . |
|       +---+       |               |
| .   . | . | .   . | .   .   .   . |
+-------+   +-------+-----------+   |
| .   .   .   .   . | .   .   . | . |
|   +---+   +-------+           +---+
| . | . | . | .   . | .   .   . | . |
|   |   +---+   +---+       +---+   |
| . | .   .   . | . | .   . | .   . |
+---+       +---+   +---+   |       |
| . | .   . | .   .   . | . | .   . |
|   +---+   +---+       +---+       |
| .   . | . | . | .   . | .   .   . |
|       +---+   +---+   +-------+   |
| .   . | .   .   . | .   .   . | . |
|       +-------+   +-----------+---+
| .   .   .   . | .   .   .   .   . |
+---------------+-------------------+
$\endgroup$
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  • $\begingroup$ In a comment below, key references were given. The law of leftovers (LoL; sudopedia.org/wiki/Law_of_Leftovers) was used to identify invalid Jigsaw Sudoku Layout. $\endgroup$
    – JCO
    Mar 4 at 8:12

1 Answer 1

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The given jigsaw layout may be proved invalid using a solving technique I learned by watching Cracking the Cryptic.

Consider columns 6 through 9, marked in blue. These contain all of the digits from 1 to 9, four times. Now consider the three regions to the right, marked in red. These are three sets of the digits 1 to 9. So the difference between them, marked in purple, must be one set of the digits 1 to 9. An application of set theory to prove the rules of Sudoku are violated

The purple set spans two different regions at the bottom of the grid. The five cells in the upper of the two regions, marked in lime and filled in with A, must appear in the lower of the two regions. Because the purple region is a complete set of digits, the lime digits cannot appear in the lower four purple cells, so they must appear in the cells marked in brown, also filled in with A.

The placeholder A represents one of the five digits which appear in the lime region. There are now six As on row eight, and only five digits to distribute among them. This would require one of the digits to repeat, which violates the rule of Sudoku that no digit may repeat in a row.

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  • 1
    $\begingroup$ Nice solution! Thanks for solving the puzzle! A reference on this subject will be added later (as a comment). Since alternative solutions may still appear, I will postpone that comment. $\endgroup$
    – JCO
    Feb 29 at 21:49
  • 2
    $\begingroup$ @Oliphaunt there is no requirement to do that at all, the OP can accept the answer whenever. It's completely up to them $\endgroup$ Mar 1 at 2:52
  • 1
    $\begingroup$ @Oliphaunt I think not accepting the answer immediately can have its benifits in terms of rep points for both the op and answerer as it still draws people into the puzzle, as users will go ‘o nice an unsolved problem let me have a go’ only to be effectively clickbaited but I don’t see anything wrong with immediately accepting the answer that being said. $\endgroup$
    – PDT
    Mar 1 at 7:46
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    $\begingroup$ @Oliphaunt: The main benefit of not accepting an answer is that new better answers may pop up. This is particularly true on Stack Overflow where beginners are tempted to accept the first answer that "solves" their problem, but generally are not in a position to judge the quality of the answers, and the first answer may be of particularly poor quality. On Puzzling... I don't know. And this solution looks very good already: short, clear diagrams, clear explanations. It'd be tough to beat. $\endgroup$ Mar 1 at 9:12
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    $\begingroup$ This question was 1st posed (sudokuxtra.com/forum/read.php?2,759) and subsequently pursued by Mathimagics. Key links: (forum.enjoysudoku.com/…) and (forum.enjoysudoku.com/…). Mathimatics approach was computational. I've recently become interested in this matter from the viewpoint of a manual solver. I had the help of an extremely strong sudoku player to analyse the hardest puzzles shown in that links. The above puzzle is a sample that I created to introduce the subject. $\endgroup$
    – JCO
    Mar 1 at 22:41

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