Abstract: We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed by mutation from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of (derived) monodromy groups of associated families of Calabi-Yau varieties. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new and more informative proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. This is joint work in progress with Hugh Thomas.