Is 1100 a valid state for this machine? A room starts out empty. Every hour, either 2 people enter or 4 people leave. In exactly a year, can there be exactly 1100 people in the room?
I think there can be because 1100 is even, but how do I prove/disprove it?

Exactly a year is $24*365=8760$ hours (or $8784$ for a leap year-maybe you need to try both).  If there are $x$ times that two people enter and $y$ times that four people leave, you want $x+y=8760,\ 2x-4y=1100$.  Two equations in two unknowns, and if the solution is integral you are there.
