<s>[INST] Necessary conditions for a Sudoku puzzle to have no repetitions Is it true that if a Sudoku puzzle has the following features there will be no repetitions in rows, columns and $3 \times 3$ subsquares?


*

*The sum of each row must be $45$

*The sum of each column must be $45$

*The sum of each $3 \times 3$ subsquare must be $45$


If so, why? Is there a mathematical proof? If not, why? Is there a case where these conditions are satisfied, but is there at least one repetition?
Thanks!
 [/INST] No. For instance, this "sudoku" fulfills your conditions, but has some repetitions:
$$
\begin{array}{|ccc|ccc|ccc|}
\hline 5&5&5&5&5&5&5&5&5\\
 5&5&5&5&5&5&5&5&5\\
 5&5&5&5&5&5&5&5&5\\
\hline 5&5&5&5&5&5&5&5&5\\
 5&5&5&5&5&5&5&5&5\\
 5&5&5&5&5&5&5&5&5\\
\hline 5&5&5&5&5&5&5&5&5\\
 5&5&5&5&5&5&5&5&5\\
 5&5&5&5&5&5&5&5&5\\
\hline
\end{array}
$$
</s>