Using standard perturbation theory, simple new double and triple summation expressions are obtained.

It is well known that for each n ≡ 1 or 3 (mod 6) there is a planar Steiner quasigroup (briefly, squag) of cardinality n (Doyen (1969) and Quackenbush (1976)). A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element subsquag (Quackenbush (1976)). In fact, Quakenbush has stated that there should be such semi-planar squags. In this paper, we construct a s...