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Context. The spatial distribution of molecules around starless cores is a powerful tool for studying the physics and chemistry governing the earliest stages of star formation. Aims. Our aim is to study the chemical differentiation in starless cores to determine the influence of large-scale effects on the spatial distribution of molecules within the cores. Furthermore, we want to put observational constraints on the mechanisms responsible in starless cores for the desorption of methanol from the surface of dust grains where it is efficiently produced. Methods. We mapped methanol, CH3OH, and cyclopropenylidene, c-C3H2, with the IRAM 30m telescope in the 3 mm band towards six starless cores embedded in different environments, and in different evolutionary stages. Furthermore, we searched for correlations among physical properties of the cores and the methanol distribution. Results. From our maps we can infer that the chemical segregation between CH3OH and c-C3H2 is driven by uneven illumination from the interstellar radiation field (ISRF). The side of the core that is more illuminated has more C atoms in the gas-phase and the formation of carbon-chain molecules like c-C3H2 is enhanced. Instead, on the side that is less exposed to the ISRF the C atoms are mostly locked in carbon monoxide, CO, the precursor of methanol. Conclusions. We conclude that large-scale effects have a direct impact on the chemical segregation that we can observe at core scale. However, the non-thermal mechanisms responsible for the desorption of methanol in starless cores do not show any dependency on the H2 column density at the methanol peak.