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.