Proof for Ramanujan's Oblong magic square

Question

In his first notebook, Ramanujan discusses a $3 X 4$ magic square which he calls oblongs. In this he suggests that following would be the elements of a magic square: $$ \begin{array} {|r|r|r|} \hline A& C+D&A+2D&C+3D \\ \hline B+6D& B+4D& B+2D& B \\ \hline C& A+D& C+2D& A+3D \\ \hline \end{array}$$

where $A,B,C,D$ are all positive integers.I am not able to get the proof behind this solution.

Kindly give suggestions as to how I can prove this.

Answer 1

You can't prove it because it isn't true for general whole number values of $A,B,C,D$. The columns all properly sum to $A+B+C+6D$, but two of the rows sum to $2A+C+6D$ and the other row sums to $4B+12D$. Presumably Ramanujan was saying there exist some integers that render the row sums equal and the oblong fully magic.

By the way, we cannot ever form a magic oblong with consecutive whole numbers unless the length and width of the rectangle are either both even or both odd.

Answer 2

Just to show what we are talking about (which has -for whatever reason - not been shown in the OP). Here a screenshot from BC Berndt's collection of Ramanujan's first notebook (pg. 20):

(Bruce C. Berndt: Ramanujan's Notebooks, Part I, Springer, 1985)