  Consider a complete communication network on $n$ nodes, each of which is a
state machine. In synchronous 2-counting, the nodes receive a common clock
pulse and they have to agree on which pulses are "odd" and which are "even". We
require that the solution is self-stabilising (reaching the correct operation
from any initial state) and it tolerates $f$ Byzantine failures (nodes that
send arbitrary misinformation). Prior algorithms are expensive to implement in
hardware: they require a source of random bits or a large number of states.
  This work consists of two parts. In the first part, we use computational
techniques (often known as synthesis) to construct very compact deterministic
algorithms for the first non-trivial case of $f = 1$. While no algorithm exists
for $n < 4$, we show that as few as 3 states per node are sufficient for all
values $n \ge 4$. Moreover, the problem cannot be solved with only 2 states per
node for $n = 4$, but there is a 2-state solution for all values $n \ge 6$.
  In the second part, we develop and compare two different approaches for
synthesising synchronous counting algorithms. Both approaches are based on
casting the synthesis problem as a propositional satisfiability (SAT) problem
and employing modern SAT-solvers. The difference lies in how to solve the SAT
problem: either in a direct fashion, or incrementally within a counter-example
guided abstraction refinement loop. Empirical results suggest that the former
technique is more efficient if we want to synthesise time-optimal algorithms,
while the latter technique discovers non-optimal algorithms more quickly.
