<s>[INST] Can a solved Sudoku game have an invalid region if all rows and columns are valid? 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 $1-9$)?
 [/INST] 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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