Hello,
When I put a cigarette in the toilet and then flush the water, the
cigarette stays there. Most other things do flush, but not the
cigarette...
I was just wondering why that is.

Try this.. throw a bit of thermocole [The white packing material] into
the toilet & flush it... it wont get flushed either... thats because
being less dense very lighter than water, and as they contain air
within them which creates additional bouyancy [the cigarette bit you
are throwing is the filter, which is porous & has air in it] , both
the cigarette butt & the thermocole bit will not get pushed down into
the water long enough to actually get flushed.

Just to add.. the cigarette butt, if left in the toilet bowl for a
while, will eventually absorb water and become enough heavier , so as
to get flushed.

Yeah, same trouble here in the southern hemisphere. They become soggy
and difficult to relight also.