GEOMETRICAL constraints, first identified by Euler, dictate that all of the closed carbon cages known as fullerenes must contain twelve pentagonal rings of carbon atoms1. In all of the fullerenes synthesized so far, each pentagon is surrounded by hexagonal rings2. Indeed, this has been proposed as a criterion for fullerene stability-the 'isolated-pentagon rule'1,3-on the basis that adjacent pentagons are expected to be chemically reactive. Buck-minsterfullerene (C60) is the smallest fullerene for which the isolated-pentagon rule can be satisfied; smaller, adjacent-pentagon fullerenes have not been formed in bulk, but have been identified previously as cations4-6. Here we report experimental evidence for the heightened chemical reactivity of cations of the adjacent-pentagon fullerenes C56 and C58, relative to C60x+, which provides support for the basic assumptions underlying the isolated-pentagon rule. Our findings suggest that, if fullerenes such as C56 and C58 are produced as intermediates or byproducts of C60 generation either in the laboratory or in natural environments, they should form derivatives readily.