Question

[For Questions 1 & 2]

There was once a crooked but witty man Douglas charged for the crime of
felony. He was kept in a prison cell which was guarded by a hefty officer. The
cell was situated at the beginning of a long straight corridor partitioned by five
doors. The doors operated on different time switches so that the first, which
separated the cell from the corridor, opened every 1 minute 45 seconds, the
second every 1 minute 10 seconds, the third every 2 minutes 55 seconds, the
fourth every 2 minutes 20 seconds, and the fifth, which was at the end of the
corridor, every 35 seconds. Every once in a while, the five doors opened
simultaneously. When this happened, the guard arrived, looked down the
corridor to check the cell, and then left. Douglas calculated that in making his
escape it would take 20 seconds to cover the distance between consecutive
doors, which was longer than the amount of time a door stayed open. He also
knew that if he stayed in the corridor for longer than two and a half minutes, at a
stretch, an alarm would sound. So he had to escape in the shortest possible time.
Given that Douglas was smart enough to keep the track of all time.
Question 1:How much time had already passed when Douglas started
moving?
A. 18m 40sec B. 19m 15sec C 19m 50sec D. Prisoner cannot escape
Question 2:How long before the guard returned does Douglas cleared the
last door?
12m 50sec
B 13m 25sc
D. Douglas

Answer 1

B. 19 min 15 sec

B. 13 min 25 sec

Explanation:

Door 1 opens every 1 min 45 sec, or 105 sec.

Door 2 opens every 1 min 10 sec, or 70 sec.

Door 3 opens every 2 min 55 sec, or 175 sec.

Door 4 opens every 2 min 20 sec, or 140 sec.

Door 5 opens every 35 sec.

The greatest common factor is 35 seconds, so we can measure the time in units of 35 seconds.

Door 1 opens every 3 units.

Door 2 opens every 2 units.

Door 3 opens every 5 units.

Door 4 opens every 4 units.

Door 5 opens every 1 unit.

The least common multiple of 3, 2, 5, 4, and 1, is 60. So every 60 units, all five doors will open, and the guard will look down the corridor to check on the prisoner. Douglas must escape before this time.

In order to escape in the shortest time possible, Douglas should time his escape so that each door opens 1 unit after the door before it. It takes Douglas 20 seconds to move from one door to another, so he will have enough time to get to the next door before it opens.

Let's say Douglas starts moving when Door 1 opens for the nth time. In other words, 3n units have passed before he starts moving. That means Door 2 should open after 3n + 1 units. Door 3 should open after 3n + 2 units. Door 4 should open after 3n + 3 units. And Door 5 should open after 3n + 4 units.

Since Door 2 opens every 2 units, 3n + 1 should be a multiple of 2.

Since Door 3 opens every 5 units, 3n + 2 should be a multiple of 5.

Since Door 4 opens every 4 units, 3n + 3 should be a multiple of 4.

Since Door 5 opens every 1 unit, 3n + 4 should be a multiple of 1.

By trial and error, n = 11.

So Douglas starts moving after 33 units, or 1155 seconds, or 19 min 15 sec.

Douglas clears the fifth door after 37 units, which leaves 23 units to spare, or 805 seconds, or 13 min 25 sec.