I'm getting fed up with the rat, and I wonder if there is some way to deal with it or if the constant frustration is meant to be an unavoidable part of the game.
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Whenever I try to poison it, it just eats the plank and bites me, with no apparent ill effect of the poison.
Yes, it's possible to get rid of the rat, but...
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it will require several doses of poison
You can also limit the the disturbances by...
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boarding up the hole, using earplugs or sleep medicine, or leaving the lights on
Have you managed to explore other areas in the game before getting tired or sick? I am stuck at the beginning of the game with trying to tackle the exam, have a date with a student, needing to finish job to make money, getting tired or sick, trying to sleep, etc.