  The concept of a k-translatable groupoid is introduced. Those k-translatable
quadratical quasigroups induced by the additive group of integers modulo m,
where k<40, are listed for m<1200. The fine structure of quadratical
quasigroups is explored in detail and the Cayley tables of quadratical
quasigroups of orders 5, 9, 13 and 17 are produced. All but those of order 9
are k-translatable, for some k. Open questions and thoughts about future
research in this area are given.
