The Hofstadter butterfly, a plot of the band spectra of almost Mathieu operators at rational frequencies, has become a pictorial symbol of the field of quasiperiodic operators and has gained renewed prominence through experimental study of moire materials. It is visually clear from this plot that for all irrational frequencies the spectrum must be a Cantor set, a statement that has been dubbed the ten martini problem. It has been established for the almost Mathieu operators, exploiting various special features of this family, in a work that has become a part of Artur Avila's Fields medal citation. We will discuss a recently developed robust method allowing to establish it for a large class of one-frequency quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families. The proof builds on the recently developed duality approach to Avila's global theory and partial hyperbolicity of the dual cocycles. The talk is based on work joint with L. Ge, J. You, and Q. Zhou, and is aimed at the general audience.