^{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…!
}

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

*Your challenge*, therefore, is as follows:

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

*or*

**B)** Prove that no such square exists.

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.