  The Tarski number of a non-amenable group G is the minimal number of pieces
in a paradoxical decomposition of G. In this paper we investigate how Tarski
numbers may change under various group-theoretic operations. Using these
estimates and known properties of Golod-Shafarevich groups, we show that the
Tarski numbers of 2-generated non-amenable groups can be arbitrarily large. We
also use the cost of group actions to show that there exist groups with Tarski
numbers 5 and 6. These provide the first examples of non-amenable groups
without free subgroups whose Tarski number has been computed precisely.
