Question 11
A collection of $8$ cubes consists of one cube with edge-length $k$ for each integer $k,$ $1 \le k \le 8.$ A tower is to be built using all $8$ cubes according to the rules:
• Any cube may be the bottom cube in the tower.
• The cube immediately on top of a cube with edge-length $k$ must have edge-length 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 $1000?$
• Any cube may be the bottom cube in the tower.
• The cube immediately on top of a cube with edge-length $k$ must have edge-length 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 $1000?$