Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")?
The diagram on the left is intended as a general representation. On the right, I created a $5 \times 5$ multiplication table (entries omitted) and summed them in their corresponding color. The width of the border was $k = 1$, and I found $\sum$frame $= 144$ but $\sum$picture $= 81$. So, in this case, the sums were not equal.
(Yes, there is a quick parity argument to show the sums here aren't equal, but I wanted to carry out the computations explicitly in an example to ensure that my question is understood.)
If the answer is no such table exists: how do you prove this?
If the answer is yes: what is the minimal $n$ (and its minimal $k$) for which this is possible?