The hadronic decay ηc(2S)→3(π+π-) is observed with a statistical significance of 9.3 standard deviations using (448.1±2.9)×106 ψ(3686) events collected by the BESIII detector at the BEPCII collider. The measured mass and width of ηc(2S) are (3643.4±2.3 (stat)±4.4 (syst)) MeV/c2 and (19.8±3.9 (stat)±3.1 (syst)) MeV, respectively, which are consistent with the world average values within two standard deviations. The product branching fraction B[ψ(3686)→γηc(2S)]×B[ηc(2S)→3(π+π-)] is measured to be (9.2±1.0 (stat)±1.2 (syst))×10-6. Using B[ψ(3686)→γηc(2S)]=(7.0-2.5+3.4)×10-4, we obtain B[ηc(2S)→3(π+π-)]=(1.31±0.15 (stat)±0.17 (syst) (+0.64-0.47) (extr))×10-2, where the third uncertainty is from B[ψ(3686)→γηc(2S)]. We also measure the χcJ→3(π+π-) (J=0, 1, 2) decays via ψ′→γχcJ transitions. The branching fractions are B[χc0→3(π+π-)]=(2.080±0.006 (stat)±0.068 (syst))×10-2, B[χc1→3(π+π-)]=(1.092±0.004 (stat)±0.035 (syst))×10-2, and B[χc2→3(π+π-)]=(1.565±0.005 (stat)±0.048 (syst))×10-2. © 2022 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/"Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP3.