Freaks City

We examine the problem of merging \(n\) initial unit vectors \(\mathbf{e}_1,\ldots,\mathbf{e}_n\) via operations that combine two vectors \(\mathbf{u},\mathbf{v}\) into either \(2\mathbf{u} - \mathbf{v}\) or \(2\mathbf{v} - \mathbf{u}\), depending on which one "wins". Each final vector’s \(i\)-th component is of the form \((-1)^{c_i}2 ...