ABSTRACT. We give lower bounds in Resolution-with-bounded-conjunction, Res$(s)$, for families of contradictions where witnesses are given in the unusual binary encodings. The two families we focus on are the $k$-Clique Formulas and those associated with the (weak) Pigeonhole Principle. If one could give lower bounds in Res$(\log$) for such families under the binary encoding, then these would translate to lower bounds for the more typical unary encoding in Resolution, Res$(1)$. Such a lower bound is not possible for certain very weak Pigeonhole Principles, but might be dreamt of for the $k$-Clique Formulas.