mathematics - Unsolved Mysteries: Magic Square of Squares

^{This is the first in what will hopefully be a series of Unsolved Mysteries posts.}

^{Note that this puzzle has no known solution, nor any proof that a solution is impossible. We will see how smart the denizens of Puzzling.SE actually are...!}

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Most people are familiar with the concept of a Magic Square. (If not, follow the link to read up on it.)

There are algorithms available that make it trivial to construct a magic square of almost any size, but by adding a few constraints to the problem, it becomes much more challenging.

Consider the following $4\times4$ magic square, where every entry is itself a square number, and the rows, columns and diagonals all sum to $8515$:

$$ \begin{array}\\ 68^2&29^2&41^2&37^2\\ 17^2&31^2&79^2&32^2\\ 59^2&28^2&23^2&61^2\\ 11^2&77^2&8^2&49^2 \end{array} $$

Note that

$68^2 + 29^2 + 41^2 + 37^2 = 17^2 + 31^2 + 79^2 + 32^2$
but
$68 + 29 + 41 + 37 \ne 17 + 31 + 79 + 32$

Only the squared values have the properties of a magic square.

Many such $4\times4$ squares have been constructed, but as of yet, no one has succeeded in constructing a $3\times3$ magic square with the same property, nor in proving that no such magic square exists.

Your challenge, therefore, is as follows:

A) Build a $3\times3$ magic square where each of the nine entries in the square is itself a square number.

or

B) Prove that no such square exists.

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For the pedantic among us (you know who you are), here are a few additional constraints:

• Each entry in the square must be unique. (A square consisting entirely of $4$s is not valid.)


• The definition of "square number" implies this, but I will spell it out here for those who like to quibble: The entries (before squaring) must be integers. Thus a magic square using values $\{ \sqrt1^2, \sqrt2^2, \sqrt3^2, \sqrt4^2, \sqrt5^2, \sqrt6^2, \sqrt7^2, \sqrt8^2, \sqrt9^2\}$ is not valid (although, of course, $\sqrt1^2$, $\sqrt4^2$, and $\sqrt9^2$ can be used in a square, being proper square numbers ($=1^2, 2^2, 3^2$).


• This also means that using complex numbers, limits, representations of infinity, or any other abstract mathematical concept is not valid. The intent of the question is obvious; please stick to that.