For odd $$$n$$$, filling the $$$i^{th}$$$ row with number $$$i$$$ will give us the satisfied matrix.

For $$$n=2$$$, we can show that there is no satisfied matrix by simply bruteforce.

For $$$n=4 \times k$$$, the satisfied matrix is constructed by:

• filling rows $$$i = 1 \rightarrow k$$$ with $$$2 \times i - 1$$$.
• filling rows $$$i = k + 1 \rightarrow 3 \times k$$$ with $$$(4k, 4k-2, 4k-4, ..., 2, 2, 4, 6, ..., 4k)$$$.
• filling rows $$$i=3 \times k + 1 \rightarrow 4 \times k$$$ with $$$2 \times i - 4 \times k - 1$$$

For $$$n=4 \times k + 2$$$, the satisfied matrix is constructed by:

• filling rows $$$i = 1$$$ and $$$i = 3 \rightarrow 2 \times k + 2$$$ with $$$(2, 4, ..., 4k, 4k+2, 4k+2, 1, 3, …, 2k−1, 2k+3, …, 4k+1)$$$.
• filling rows $$$i = 2$$$ and $$$i = 2 \times k + 3 \rightarrow 4 \times k + 2$$$ with $$$(2, 4, ..., 4k, 2k+1, 2k+1, 1, 3, …, 2k−1, 2k+3, …, 4k+1)$$$.