This problem solving activity has a number focus.
A prisoner sits in his cell planning his escape. The prisoner is kept in by 5 laser beams, which operate along a corridor. Each laser is switched off at a specific time interval for just long enough to allow a person to walk through. The time between being switched off for each laser is shown below:
Laser One = every 3 minutes
Laser Two = every 2 minutes
Laser Three = every 5 minutes
Laser Four = every 4 minutes
Laser Five = every 1 minutes
The guard patrols and checks the prisoner each time all the laser beams are off simultaneously. Because each laser only switches off for a short time the prisoner knows he can only get past one laser at a time. He has to get past the five lasers from 1 to 5 in order. Laser One is at the entrance of the prisoner’s cell and laser Five is at the door to the outside. He also knows that if he spends longer than 4 minutes 12 seconds in the corridor an alarm will go off.
Can the prisoner escape without the alarm in the corridor going off?
If he can escape, how many minutes should he wait before passing Laser One?
How much time will he have after passing Laser Five before the guard raises the alarm?
This problem requires students to fully understand what the problem is asking, to use logic, and to find the lowest common multiple of the numbers 1, 2, 3, 4, and 5. Students should possess fluent knowledge of basic multiplication facts, and should have experience with multi-step problems, to work successfully with this problem.
A prisoner sits in his cell planning his escape. The prisoner is kept in by 5 laser beams, which operate along a corridor. Each laser is switched off at a specific time interval for just long enough to allow a person to walk through. The time between being switched off for each laser is shown below:
Laser One = every 3 minutes
Laser Two = every 2 minutes
Laser Three = every 5 minutes
Laser Four = every 4 minutes
Laser Five = every 1 minutes
The guard patrols and checks the prisoner each time all the laser beams are off simultaneously. Because each laser only switches off for a short time the prisoner knows he can only get past one laser at a time. He has to get past the five lasers from 1 to 5 in order. Laser One is at the entrance of the prisoner’s cell and Laser Five is at the door to the outside. He also knows that if he spends longer than 4 minutes 12 seconds in the corridor an alarm will go off.
Can the prisoner escape without the alarm in the corridor going off?
If he can escape, how many minutes should he wait before passing Laser One?
How much time will he have after passing Laser Five before the guard raises the alarm?
Can the groups devise a similar problem using different time sequences? (Use times in either minutes or minutes and seconds.)
The prisoner can escape and will have 23 minutes before the guard will sound the alarm.
When the guard leaves then let that time equal 0 units’ time. Each laser turns off in a sequence of 1, 2, 3, 4, and 5 minutes. The guard will return every 60 minutes. This is the lowest common multiple of 1, 2, 3, 4 and 5.
The prisoner can only spend 4 minutes 12 seconds in the corridor. This means that the prisoner must wait for a sequence when all the laser beams go off one after the other. The only five successive times possible between 0 and 60 minutes are 33, 34, 35, 36 and 37.
This can be found by listing the laser times.
Laser 1 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33...
Laser 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, ...
Laser 3 = 5, 10, 15, 20, 25, 30, 35, ...
Laser 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Laser 5 = 1, 2, 3, ..., 35, 36, 37..
Or realising you are looking for a sequence of numbers where the first number is a multiple of 3, the second is an even number, the third is a multiple of 5, the fourth is a multiple of 4, and the fifth can be any number as it is a 1s number.
The prisoner must therefore wait until 33 minutes after the guard leaves before entering the corridor. He clears the final door 23 minutes before the guard returns.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/escape at 8:57pm on the 26th February 2024