A Solution to the

Elevator Problem

Problem:

Two elevators leave the nth floor at 2:00 P.M. The faster elevator takes one minute to
travel between floors and the slower elevator takes two minutes to travel between floors.
The first elevator to reach a floor must stop for three minutes to take on passengers. If
both elevators arrive at a floor at the exact same time, they become confused and do
not stop for passengers.

If the final stop for an elevator is the lobby (1st floor), then describe n if the faster
elevator arrives at the lobby first. Describe n if the slower elevator arrives at the lobby
first.

(Source: Adapted from Mathematics Teaching in the Middle School, Apr 1994).

Analysis:

Generate a table to describe the total time it takes each elevator to move after floors. If
you vary the starting position (cell B1), then the spreadsheet is a powerful way to see the
pattern in the solutions.

The highlighted cells represent an elevator that has waited three minutes for passengers
to load.

Summary of Analysis:

A quick way to determine an answer is the slower elevator will reach the ground first if the
starting floor is a multiple of three; otherwise, the faster elevator will reach the ground
first.

Notice the geometric pattern in this amazingly slow elevator. The faster elevator appears
to be stopping twice as often as the slower elevator. The faster elevator will stop after its
1st, 3rd, 4th, 6th, 7th,...etc floors, or the union of the sequences 1 + 3n, and 3n, where
n is a positive integer. Thus, the faster elevator will reach the ground first if the number
of floors minus one MOD 3 = 0 or 1, or the starting floor MOD 3 = 1 or 2. In other words,
the faster elevator will reach the first floor first if the starting floor minus one is divisible
by any number in the sequence 1 + 3n, or 3n. For example, if you start on the 65th
floor, then you will be traveling 64 floors; 64 is in the sequence 1 + 3n between 61 and
67, so the faster elevator will reach the ground first.

On the other hand, the slower elevator reachers the ground first if it has traveled 2
floors, 5 floors, 8 floors, etc, or by the sequence 3n + 2, where n is any positive integer.
Thus, the slower elevator will reach the ground first if the number of floors minus one
MOD 3 = 2, or the starting floor MOD 3 = 0. For example, if you start on the 48th floor,
then you will be traveling 47 floors; 47 is in the sequence between 44 and 50, so the
slower elevator will reach the ground first.