A game concerning Sudoku We (me and you) start with a blank $9\times 9$ blank square (as an empty Sudoku) and I fill the first three rows legally according to Sudoku rules. Is it always possible that you complete the Sudoku with this given three rows? 
 A: Suppose you filled the first three rows according to Sudoku rules, let a1, a2, a3 be the 3x1 vectors in the top-left block; b1, b2, b3 be the next three 3x1 vectors in the top-middle block; and c1, c2, c3 be the last three 3x1 vectors in the top-right block.
Then consider:
\begin{array}{|ccc|ccc|ccc|}
\hline | & | & | & | & | & | & | & | & |\\
 {\vec{a}_{1}} &  {\vec{a}_{2}} &  {\vec{a}_{3}} &  {\vec{b}_{1}} &  {\vec{b}_{2}} &  {\vec{b}_{3}} &  {\vec{c}_{1}} &  {\vec{c}_{2}} &  {\vec{c}_{3}}\\
| & | & | & | & | & | & | & | & |\\
\hline  &  &  &  &  &  &  &  & \\
 & ? &  &  & ? &  &  & ? & \\
 &  &  &  &  &  &  &  & \\
\hline  &  &  &  &  &  &  &  & \\
 & ? &  &  & ? &  &  & ? & \\
 &  &  &  &  &  &  &  & \\
\hline \end{array} and fill it with\begin{array}{|ccc|ccc|ccc|}
\hline | & | & | & | & | & | & | & | & |\\
 {\vec{a}_{1}} &  {\vec{a}_{2}} &  {\vec{a}_{3}} &  {\vec{b}_{1}} &  {\vec{b}_{2}} &  {\vec{b}_{3}} &  {\vec{c}_{1}} &  {\vec{c}_{2}} &  {\vec{c}_{3}}\\
| & | & | & | & | & | & | & | & |\\
\hline | & | & | & | & | & | & | & | & |\\
 {\vec{a}_{2}} &  {\vec{a}_{3}} &  {\vec{a}_{1}} &  {\vec{b}_{2}} &  {\vec{b}_{3}} &  {\vec{b}_{1}} &  {\vec{c}_{2}} &  {\vec{c}_{3}} &  {\vec{c}_{1}}\\
| & | & | & | & | & | & | & | & |\\
\hline | & | & | & | & | & | & | & | & |\\
 {\vec{a}_{3}} &  {\vec{a}_{1}} &  {\vec{a}_{2}} &  {\vec{b}_{3}} &  {\vec{b}_{1}} &  {\vec{b}_{2}} &  {\vec{c}_{3}} &  {\vec{c}_{1}} &  {\vec{c}_{2}}\\
| & | & | & | & | & | & | & | & |
\\\hline \end{array} 
Note the rest of the blocks are just permutations of the columns of the blocks above. It is done in a way so that in each block, we have all 1-9 numbers, since the first three rows are Sudoku-satisfied. And each column will also have 1-9, because we used the top three blocks to make the columns. Lastly, the rows are all satisfied with 1-9, because they are just the same rows as the first three rows except permuted again.
By way of an example, consider this:
\begin{array}{|ccc|ccc|ccc|}
\hline 5 & 3 & 4 & 6 & 7 & 8 & 9 & 1 & 2\\
6 & 7 & 2 & 1 & 9 & 5 & 3 & 4 & 8\\
1 & 9 & 8 & 3 & 4 & 2 & 5 & 6 & 7\\
\hline  &  &  &  &  &  &  &  & \\
 & ? &  &  & ? &  &  & ? & \\
 &  &  &  &  &  &  &  & \\
\hline  &  &  &  &  &  &  &  & \\
 & ? &  &  & ? &  &  & ? & \\
 &  &  &  &  &  &  &  & \\
\hline \end{array} 
We will fill the remaining blocks by permuting the columns of the 3x3 blocks directly above it cyclically: 
\begin{array}{|ccc|ccc|ccc|}
\hline 5 & 3 & 4 & 6 & 7 & 8 & 9 & 1 & 2\\
6 & 7 & 2 & 1 & 9 & 5 & 3 & 4 & 8\\
1 & 9 & 8 & 3 & 4 & 2 & 5 & 6 & 7\\
\hline 3 & 4 & 5 & 7 & 8 & 6 & 1 & 2 & 9\\
7 & 2 & 6 & 9 & 5 & 1 & 4 & 8 & 3\\
9 & 8 & 1 & 4 & 2 & 3 & 6 & 7 & 5\\
\hline 4 & 5 & 3 & 8 & 6 & 7 & 2 & 9 & 1\\
2 & 6 & 7 & 5 & 1 & 9 & 8 & 3 & 4\\
8 & 1 & 9 & 2 & 3 & 4 & 7 & 5 & 6
\\\hline \end{array} 
