In a Wyner model each link only interferes with links adjacent to it. We consider the problem of finding the optimal power allocation which maximizes the sum-rate of such a network. Each link has a maximum power constraint and the power allocation is assumed to be time and frequency flat. In the case of 3- and 4-link Wyner models, we show that the optimal power schemes are in fact binary, i.e. a link is either switched off or turned on at full power. The problem is then extended to larger-sized Wyner models by limiting to optimal binary power schemes. Interesting phase transitions are observed as the interference cross-gain, ε, traverses various thresholds.