There is a well known Ramanujan's magic square of the 4th order (see Figure 1.). The underlying structure of Ramanujan's magic square is given in Figure 2.
$$\begin{array}{ |c|c|c|c| } \hline 22 & 12 & 18 & 87 \\ \hline 88 & 17 & 9 & 25 \\ \hline 10 & 24 & 89 & 16 \\ \hline 19 & 86 & 23 & 11 \\ \hline \end{array}$$ $$\text{Figure 1.}$$
$$\begin{array}{ |c|c|c|c| } \hline \text{A} & \text{B} & \text{C} & \text{D} \\ \hline \text{D}+1 & \text{C}-1 & \text{B}-3 & \text{A}+3 \\ \hline \text{B}-2 & \text{A}+2 & \text{D}+2 & \text{C}-2 \\ \hline \text{C}+1 & \text{D}-1 & \text{A}+1 & \text{B}-1 \\ \hline \end{array}$$ $$\text{Figure 2.}$$
I have managed to create variation of Ramanujan's magic square , where date of the birth is given in the main diagonal rather than in the first row (see Figure 3.) .The underlying structure of this variation is given in Figure 4.
$$\begin{array}{ |c|c|c|c| } \hline 22 & 88 & 10 & 19 \\ \hline 17 & 12 & 86 & 24 \\ \hline 89 & 23 & 18 & 9 \\ \hline 11 & 16 & 25 & 87 \\ \hline \end{array}$$ $$\text{Figure 3.}$$
$$\begin{array}{ |c|c|c|c| } \hline \text{A} & \text{D}+1 & \text{B}-2 & \text{C}+1 \\ \hline \text{C}-1 & \text{B} & \text{D}-1 & \text{A}+2 \\ \hline \text{D}+2 & \text{A}+1 & \text{C} & \text{B}-3 \\ \hline \text{B}-1 & \text{C}-2 & \text{A}+3 & \text{D} \\ \hline \end{array}$$ $$\text{Figure 4.}$$
I was using trial and error method, but I am interested if there are some more concise methods for creating nontrivial variations (i.e. variations where birthdate isn't given in the first or in the fourth column, in the fourth row , etc.) of the Ramanujan's magic square?
P.S.
You can generate both types of your personal magic squares here.