A collection of 8 cubes consists of one cube with edge-length k for each integer k, 1 ≤ k ≤ 8. A tower is to be built using all 8 cubes according to the rules: 1) Any cube may be the bottom cube in the tower. 2) The cube immediately on top of a cube with an edge length of k must have an edge length of at most k+2. Let T be the number of different towers that can be constructed. What is the remainder when T is divided by 1000?

Guest Jan 7, 2023