We introduce "puzzles of quasi-finite type" which are the counterparts of our subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of combinatorial puzzles as defined in complex dynamics. We are able to analyze these dynamics defined by entropy conditions rather completely, obtaining a complete classification with respect to large entropy measures and a description of their measures with maximum entropy and periodic orbits. These results can in particular be applied to entropy-expanding maps like (x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the meromorphy of the Artin-Mazur zeta function on a large disk. This follows from a similar new result about strongly positively recurrent Markov shifts where the radius of meromorphy is lower bounded by an "entropy at infinity" of the graph.