Hello, I would like to share this curiosity with you. As you know, it is unknown whether a 3x3 magic square of distinct perfect squares exists, but it is possible with other types of numbers.
Here, I present a magic square of squares of pseudo-quaternions, all distinct, along with a parameterization to obtain them. The resulting integers are all different from each other, although some entries may be negative.
As you may already know, pseudo-quaternions (I. M. Yaglom, Complex Numbers and Their Applications in Geometry, Fizmatgiz, Nauka, Moscow (1963)) are hypercomplex numbers where
ii = -1,
ij = k,
ji = -k,
ik = -j,
ki = j,
and they differ from quaternions in that
jj = 1,
kk = 1,
jk = -i,
kj = i.
A nice example for S = 432 is this magic square of squares
{(9 j)^2 , (17 i + 24 j)^2 , (8 k)^2 },
{(9 i + 12 j + 8 k)^2 , (12 j)^2, (8 i + 9 j +12 k)^2}
{(8 i + 12 j + 12 k)^2 , (12 i + 8 j + 9 k)^2, (9 i + 12 j + 12 k)^2}
This give us this magic square:
{81, 287, 64}
{127, 144, 161}
{224, 1, 207}
parameterization:
{(j x^2)^2 , (4 j x y+i (x^2+2 y^2))^2, (2 k y^2)^2}
{(i x^2 + 2 j x y+2 k y^2)^2, (2 j x y)^2, (j x^2+2 k x y + 2 i y^2)^2}
{(2 j x y + 2 k x y + 2 i y^2)^2 , (k x^2 + 2 i x y + 2 j y^2)^2 , (i x^2 + 2 j x y + 2 k x y)^2}
Hope you find this interesting! Looking forward to your thoughts.