An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equation that determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.