  In this paper we consider the generalized Reeder's puzzle, introduced by
Reeder in 2005 and generalized by Borovoi and Evenor in 2016. We give a
detailed solution of the puzzle for the graphs of Dynkin diagrams and affine
Dynkin diagrams. We find the number of equivalence classes in each case. We
also discuss more general graphs, and prove the main theorem about graphs
(simply-laced trees) that contain E_6 as a subgraph.
