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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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By definition, if it's a solved Sudoku then all rows, columns, and regions are valid. – Peter Taylor 21 hours ago
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Given 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 oeis.org/A107739/list (number of $9 \times 9$ Sudoku grids: 6.7 sextillion), oeis.org/A002860/list (number of Latin squares: 5.5 octillion). – Tanner Swett 11 hours ago
    
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 3 hours ago
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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 23 hours ago
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@dragonfly Swap rows 2 and 4, and rows 3 and 7. The three top regions are now valid, while the rest are not. – jvdhooft 23 hours ago
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Permutating 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 23 hours ago
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Concerning 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 mathoverflow.net/q/129143 . – Emil Jeřábek 19 hours ago

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