In this paper, we study the fundamental problem of counting, which consists in computing the size of a system. We consider the distributed communication model of population protocols of finite state, anonymous and asynchronous mobile devices (agents) communicating in pairs (according to a fairness condition). This work significantly improves the previous results known for counting in this model, in terms of (exact) space complexity. We present and prove correct the first space optimal protocols solving the problem for two classical types of fairness, global and weak. Both protocols require no initialization of the counted agents.

The protocol designed for global fairness, surprisingly, uses only one bit of memory (two states) per counted agent. The protocol, functioning under weak fairness, requires the necessary log P bits (P states, per counted agent) to be able to count up to P agents. Interestingly, this protocol exploits the intriguing Gros sequence of natural numbers, which is also used in the solutions to the Chinese Rings and the Hanoi Towers puzzles.