We describe pretty examples of derived equivalences and autoequivalences
of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds
are chosen to be special, so they each have an associated K3 surface. Thus a pencil
gives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces,
and the baselocus of the original pencil—the intersection of two cubic fourfolds. They
both have crepant resolutions which are derived equivalent.