KöMaL 2007 K125

Sebastian and Vincent were building towers out of wooden building blocks. They both used identical prisms of square base. Sebastian laid the blocks on each other along their faces of smaller area, while Vincent laid them on each other along their faces of larger area. Sebastian's tower of four blocks turned out to have the same height as Vincent's tower of six blocks. Find the height of the towers, given that the sum of their base areas is 160 $cm^2$.