Since tiling patterns
,
,
,
are orthogonal with each other,
perfect-magic squares can be obtained by
where
and
Therefore, there are perfect-magic squares.
The possible combinations are given by
The number of independent perfect magic squares is given by
The factor comes from the definition in which
we take
and its transposition
is identical.
The factor
appears, since
we take
,
and its rotations
,
,
are identical.
Therefore, there exist three identical perfect-magic squares
which are
These perfect-magic squares correspond with those given by Grogono.