Multiplicative Magic Square from 9 Terms of Geometric Sequence from 1, 2, 4

Problem

 * With the first $9$ terms of the geometric sequence $1, 2, 4, \ldots$,
 * form a product of $4096$ each way.

That is, form a multiplicative magic square whose [Definition:Multiplicative Magic Constant|magic constant]] is $4096$.

Solution

 * $\begin{array}{|c|c|c|}

\hline 2 & 64 & 32 \\ \hline 256 & 16 & 1 \\ \hline 8 & 4 & 128 \\ \hline \end{array}$

Proof
Take the order $3$ (additive) magic square:

Subtract $1$ from each element:


 * $\begin{array}{|c|c|c|}

\hline 1 & 6 & 5 \\ \hline 8 & 4 & 0 \\ \hline 3 & 2 & 7 \\ \hline \end{array}$

This square is still magic.

Now replace each element $n$ with $2^n$:


 * $\begin{array}{|c|c|c|}

\hline 2^1 & 2^6 & 2^5 \\ \hline 2^8 & 2^4 & 2^0 \\ \hline 2^3 & 2^2 & 2^7 \\ \hline \end{array}$

which is:


 * $\begin{array}{|c|c|c|}

\hline 2 & 64 & 32 \\ \hline 256 & 16 & 1 \\ \hline 8 & 4 & 128 \\ \hline \end{array}$