Inelastic neutron scattering recently confirmed the theoretical prediction of a ↑↑↓↓-magnetic state along the legs of quasi-one-dimensional (quasi-1D) iron-based ladders in the orbital-selective Mott phase (OSMP). We show here that electron-doping of the OSMP induces a whole class of novel block-states with a variety of periodicities beyond the previously reported π/2 pattern. We discuss the magnetic phase diagram of the OSMP regime that could be tested by neutrons once appropriate quasi-1D quantum materials with the appropriate dopings are identified.Introduction. Competing interactions in strongly correlated electronic systems can induce novel and exotic effects. For example, in the iron-based superconductors [1][2][3][4] charge, spin, and orbital degrees of freedom combine phenomena known from cuprates with those found in manganites. Prominent among these novel effects is the orbital-selective Mott phase (OSMP) [5], where interactions acting on a multi-orbital Fermi surface cause the selective localization of electrons on one of the orbitals. As a consequence, the system is in a mixed state with coexisting metallic and Mott-insulating bands [ Fig. 1(a)]. Since the theoretical studies of the OSMP require the treatment of challenging multi-orbital models most of the investigations thus far were performed with approximations such as the the slave-particle mean field method [6][7][8][9] or dynamical mean-field theory [10][11][12]. Here, we present unambiguous numerical evidence of the OSMP within low-dimensional multi-orbital Hubbard models, unveiling a variety of new phases.The magnetic ordering associated with the OSMP could be significantly different from that observed in cuprates. The latter are described by single-band Hamiltonians and the parent compounds order in a staggered antiferromagnetic fashion. However, recent inelastic neutron scattering (INS) experiments [13] on quasi-1D ironbased materials of the 123 family (AFe 2 X 3 ; A alkali metals, X=Se,S chalcogenides) unveiled exotic π/2-block magnetic states where spins form antiferromagnetically (AFM) coupled ferromagnetic (FM) islands, in a ↑↑↓↓pattern [ Fig. 1(b)]. Similar patterns were also reported in two dimensions with iron vacancies, such as in Rb 0.89 Fe 1.58 Se 2 [14] and K 0.8 Fe 1.6 Se 2 [8,[15][16][17]. For the aforementioned compounds the OSMP state is believed to be relevant [8,13,[18][19][20].Recent theoretical investigations [21,22] showed that a multi-orbital Hubbard model in the OSMP state properly describes the INS spin spectra of π/2-block state of powder BaFe 2 Se 3 [13]. However, the origin of block magnetism and its relation to the OSMP is an intriguing and generic question that has been barely explored. We show that the block-OSMP magnetism can develop in various shapes and sizes depending on the electrondoping, far beyond the previously reported π/2-pattern. Moreover, here we develop effective Hamiltonians for the OSMP which allows for an intuitive understanding of the origin of block magnetism. Our simplified, yet accurate, model...