Presence of a distinct number line in a lattice square

Question

Fill the $n\times n$ lattice square with natural numbers $\{1,2,\cdots,n\}$ each of which is used $n$ times. Define column $j$ for a given $j$ as $(a_{i,j}), i=1,2,\cdots,n$; row $i$ for a given $j$ as $(a_{i,j}), j=1,2,\cdots,n$; positive diagonal $k$ for a given $k\ge0$ as $(a_{i,j}), j-i\equiv k(\mod n), k=0,1,2,\cdots,n-1$; negative diagonal $k$ for a given $k\le0$ as $(a_{i,j}), j+i\equiv k+2 (\mod n), k=0,1,2,\cdots,n-1$. Must there be a row, column or a diagonal (positive or negative) consists of all the distinct $n$ natural numbers?

Answer 1

Note that there are only $4n$ straight lines, and each straight line can be "blocked" by placing two copies of the same number on it; in principle that means we need only $8n$ numbers (out of the $n^2$ at our disposal) to force a counterexample. To me this suggests that the conjecture is likely to be false for all but the smallest $n$.

With this idea, it's easy to come up with a counterexample for $n=6$, and only a little harder to come up with counterexamples for $n=5$ and $n=4$: $$ \begin{pmatrix} 1&1&1&2&2&2 \\ 1&1&1&2&2&2 \\ 3&3&3&4&4&4 \\ 3&3&3&4&4&4 \\ 5&5&5&6&6&6 \\ 5&5&5&6&6&6 \end{pmatrix} \qquad \begin{pmatrix} 1&1&2&2&2 \\ 1&1&2&2&5 \\ 1&3&3&4&4 \\ 3&3&3&4&4 \\ 5&5&5&5&4 \end{pmatrix} \qquad \begin{pmatrix} 1&1&4&4 \\ 1&1&2&2 \\ 3&3&2&2 \\ 3&3&4&4 \end{pmatrix} $$

(One can show by brute force that there is no counterexample for $n=3$, even if only the main diagonals are considered.)