# 2006 AIME A Problems/Problem 11

## Problem

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 than can be constructed. What is the remainder when $T$ is divided by 1000?

## Solution

Define the sum as $x$. Notice that $a_n\ = a_{n + 3} - a_{n + 2} - a_{n + 1}$, so the sum will be: $x = (a_4 - a_3 - a_2) + (a_5 - a_4 - a_3) + \ldots (a_{30} - a_{29} - a_{28}) + a_{28}$ $x = (a_4+ a_5 \ldots a_{30}) - (a_3 + a_4 + \ldots a_{29}) - (a_2 + a_3 + \ldots a_{28}) + a_{28} + (a_1 - a_1)$

The first two groupings almost completely cancel. The third resembles $x$. $x\ = a_1 - a_3 + a_{28} + a_{30} - x$ $2x\ = a_{28} + a_{30}$ $x\ = \frac{a_{28} + a_{30}}{2}$ $a_{28}$ and $a_{30}$ are both given; the last four digits of the sum is $3668$, and half of that is $1834$. Therefore, the answer is $834$.