A central concept for random processes is mixing in its various forms. The strongest and simplest mixing condition is finite dependence, which states that variables at sufficiently well separated locations are independent. A 50-year old conundrum is to understand the relationship between finitely dependent processes and block factors (a block factor is a finite-range function of an independent family). The issue takes a very surprising new turn if we in addition impose a local constraint (such as proper coloring) on the process. In particular, this has led to the discovery of a beautiful yet mysterious random process that seemingly has no right to exist.