An extended triple system with no idempotent element (ETS) is a collection of non ordered triples of type {x,y,z} or {x,x,y} chosen from a v set in such a way that each pair (whether distinct or not) is contained in exactly one triple. (For example, in the block {x,x,y}, the pair {x,y} is said to occur one tinle.) Such a design has "'v = v(v + 3) /6 blocks and a necessary and sufficient conditi...