We describe our rediscovery of an intriguing logical game, introduced by Immerman in 1981, but then, until now, never again mentioned in the literature. The game is played on two sets A and B of structures. These multi-structural games generalize Ehrenfeucht-Fraisse games. Whereas Ehrenfeucht-Fraisse games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the r-round game if and only if there is a first-order sentence Φ with at most r quantifiers, where every structure in A satisfies Φ and no structure in B satisfies Φ. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.