Our main result is the following elementary inequality: if p and q are fixed, with \(0\leq p,q\leq 1\), then the sum of any N terms from the set \(\{p^ mq^ n:\) \(m,n=0,1,2,...\}\) does not exceed \(1+(p+q-pq)+..+(p+q- pq)^{N-1}\). From this we derive some new estimates for moment sequences. The inequality may be expressed in terms of a simple coin- tossing game. Analogous games are described, in which the coins are replaced by urns. These are more difficult to analyze and we were unable to solve them at the time of writing of the paper under review. A solution has now been found: it will appear in the author’s forthcoming article, “Double dipping: the case of the missing binomial coefficient identities.”