  We study a simple deterministic map that leads a fully connected network to
Heider balance. The map is realized by an algorithm that updates all links
synchronously in a way depending on the state of the entire network. We observe
that the probability of reaching a balanced state increases with the system
size N. Jammed states become less frequent for larger N. The algorithm
generates also limit cycles, mostly of length $2$, but also of length 3, 4, 6,
12 or 14. We give a simple argument to estimate the mean size of basins of
attraction of balanced states and discuss symmetries of the system including
the automorphism group as well as gauge invariance of triad configurations. We
argue that both the symmetries play an essential role in the occurrence of
cycles observed in the synchronous dynamics realized by the algorithm.
