FYKOS Year 25 Series 5 Problem 3 — pilgrimage of pharaohs · FYKOS 25 #25.5.3
Physics · mechanics
Problem
Mára decided to infect Aleš's four room apartment with pharaoh ants (top view of the apartment is on the picture). Ants are running all over the place but you can assume the following model of their motion. Every five minutes 60 of ants in each room moves to the neighboring rooms and the rest stay where they are now. If there is more than one neighboring room assume that the same amount of ants moved to every one of them. This process repeats itself every five minutes (yes, assume only discrete time). The ants cannot move in or out of the apartment and they are immortal.
If Mára places 1000 ants into the hallway (D) how many ants will be in each of the rooms after 5, 10 and 15 minutes? (2 body)
If at some point we found out that the distribution of ants in the rooms is NA=12, NB=25, NC=25 aND=37, how where they distributed 5 minutes earlier? (1 bod)
Bonus: How would they be distributed after essentially infinite time if we start with 1000 ants in the hallway again? Does it matter how where they distributed in the beginning? And finally - will the distribution of ants in the rooms reach a stationary value or will it oscillate? (bonus points)