Given a $9 \times 9$ solved Sudoku game with $3 \times 3$ regions, is it possible that one (or more) of the regions are invalid if all rows and columns are valid (i.e. have a unique sequence of $1-9$)?
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21By definition, if it's a solved Sudoku then all rows, columns, and regions are valid. – Peter Taylor Jun 21 '17 at 14:34
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4Given a random $9 \times 9$ Latin square (which is the same thing as a Sudoku grid filled with the digits 1 through 9 such that all rows and columns are valid), there is a 99.99988% chance that at least one of the regions will be invalid. See https://oeis.org/A107739/list (number of $9 \times 9$ Sudoku grids: 6.7 sextillion), https://oeis.org/A002860/list (number of Latin squares: 5.5 octillion). – Sophie Swett Jun 22 '17 at 00:42
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This question seems (to some extent) related: Can a sudoku with valid columns and rows be proved valid without evaluating every 3x3 inside it? – Martin Sleziak Jun 22 '17 at 08:09
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Yes, it can happen that all $3 \times 3$ regions are invalid:
\begin{array}{|ccc|ccc|ccc|} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 1 \\ 3 & 4 & 5 & 6 & 7 & 8 & 9 & 1 & 2 \\ \hline 4 & 5 & 6 & 7 & 8 & 9 & 1 & 2 & 3 \\ 5 & 6 & 7 & 8 & 9 & 1 & 2 & 3 & 4 \\ 6 & 7 & 8 & 9 & 1 & 2 & 3 & 4 & 5 \\ \hline 7 & 8 & 9 & 1 & 2 & 3 & 4 & 5 & 6 \\ 8 & 9 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 9 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \end{array}
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Thanks! Given this special case where all are invalid, is there any other case? – dragonfly Jun 21 '17 at 12:12
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6@dragonfly Swap rows 2 and 4, and rows 3 and 7. The three top regions are now valid, while the rest are not. – jvdhooft Jun 21 '17 at 12:15
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2Permutating the numbers $1$ to $9$ in this example arbitarily gives other solutions, but there will be tons of further solutions. An interesting question would be : What is the probability that a random $9\times 9$-Latin square has the desired property ? And further : What is the probability that exactly $k$ regions are invalid ($k\in [0,9])$ ? – Peter Jun 21 '17 at 12:15
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2Concerning other cases: if two regions in the same row are valid, then all three are, and dually for columns. On the other hand, any pattern of valid and invalid regions that obeys this rule is possible to realize. This, and much more, can be found at https://mathoverflow.net/q/129143 . – Emil Jeřábek Jun 21 '17 at 16:38