I filled in a $3\times \left \lfloor 1+\dfrac{n}{2}\right \rfloor$ rectangle with non negativ integers, such that the sum of the three numbers in each column is the same, and in each row all the numbers are different.
Let $k$ denote the number of columns containing the number 0.
Is it true that if $n$ is odd, then $k\leq 1$, and if $n$ is even, then $k\leq 2$?
I got an extra homework, and if I could prove the problem above, then I could solve the homework. So I need help! I am very thankful for every solution!
How about $$\begin {bmatrix} 0&1&2\\1&2&0\\2&0&1 \end {bmatrix}$$ as a counterexample for both $n=4$ and $n=5$ Why should there be any difference between even and odd $n$ because $n=2m$ and $n=2m+1$ give the same size matrix? I suspect you have left out something about $n$.