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I'm late to this thread. So, firstly, lumberjim you bastard you poisoned the order of my thoughts. I was surprised by how difficult it was to discipline my mind to actually think that one through without going through it like the question did. That was cool.
PEMDAS: The P is for Parentheses, the E is for Exponents.
With the ping-pong ball question, my immediate first thought was: "water!". When I heard that the room had no resources, I resorted to urine. The practical uses of urine that I know are: when dragging sleds with metal skates (I can't remember the proper word here) through snow, pee on the skates. I believe that the goal is to form a regular coating of ice. Urine is also a sterilizer, so you can use it to clean clothes, assuming that you have something to cover the smell. The Romans used bay leaves for this purpose.
My cheating solution to the three light bulbs question would be to watch the light under the door; but that is probably eliminated by the question.
"I have two ends, I have two rings, and a nail through the middle" -- I've been trying to figure out what this one is for the past week, rather halfheartedly.
As for the guys in the cabin, one possibility is that the hole is the chimney. They entered the cabin during the snowstorm, started a fire, and that either poisoned them, burned the cabin, or they froze to death. A simple possibility is that during the snowstorm, the roof caved in. However, I like the airplane answer better.
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