Off the top of my head: the cubes of 0,1,2,3,4,5,6,7,8 mod 9 are 0,1,-1,0,1,-1,1,-1. The sums of three of these, mod 9, are 0, 1, 2, 3, -1, -2, -3 or 0, 1, 2,

Off the top of my head: the cubes of 0,1,2,3,4,5,6,7,8 mod 9 are 0,1,-1,0,1,-1,1,-1. The sums of three of these, mod 9, are 0, 1, 2, 3, -1, -2, -3 or 0, 1, 2, 3, 6, 7, 8, so the sum of 3 cubes can not be 4 or 5 mod 9.