Entanglement is one of the most intriguing features of quantum mechanics. It is widely used in quantum communication and information processing and plays a key role in quantum computation. At the same time, entanglement is not fully understood. It is deeply rooted into the linearity of quantum theory and in the superposition principle and (for pure states) is essentially and intuitively related to the impossibility of factorizing the state of the total system in terms of states of its constituents. The characterization and quantification of entanglement is an open and challenging problem. One can give a good definition of bipartite entanglement in terms of the von Neumann entropy or the entanglement of formation. The problem of defining multipartite entanglement is more difficult and no unique definition exists. We characterize the multipartite entanglement of a system of n qubits in terms of the distribution function of the bipartite purity over balanced bipartitions. We search for maximally multipartite entangled states, whose average purity is minimal, and recast this optimization problem into a problem of statistical mechanics, by introducing a cost function, a fictitious temperature and a partition function. By investigating the high-temperature expansion, we obtain the first three moments of the distribution. We find that the problem exhibits frustration. We focus on fundamental issues and possible applications.