# Can a solved Sudoku game have an invalid region if all rows and columns are valid? [closed]

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 $1-9$)?

• By definition, if it's a solved Sudoku then all rows, columns, and regions are valid. Jun 21, 2017 at 14:34
• 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). Jun 22, 2017 at 0:42
• This question seems (to some extent) related: Can a sudoku with valid columns and rows be proved valid without evaluating every 3x3 inside it? Jun 22, 2017 at 8:09

Yes, it can happen that all $3 \times 3$ regions are invalid:
• 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 ($k\in [0,9])$ ? Jun 21, 2017 at 12:15