In class, we briefly covered what "floor" and "ceiling" mean. Very simple concepts. They were on one slide, and then we never heard about them again. But now the following homework problem has popped up:
$$\lfloor\lfloor x/2 \rfloor / 2 \rfloor = \lfloor x/4 \rfloor$$
Usually when I post a problem (especially from homework), I like to demonstrate that I'm not just asking for the answer by showing what I've done, what I know so far, and so on... but in this case, I have absolutely no idea where to even begin.
I will say that my first approach was to create a chart and try various values for "x" in order to see if there's a pattern and to make sure there was no easy counter-example to prove it false. That was all fine and well, but ultimately I couldn't figure out what to do with the results.
Googling for this is quite difficult, as everyone uses slightly different notation and therefore one search doesn't encompass all the actual results. The only clue that I've seen so far that kind of sort of makes sense, was when some guy said that "x" should be replaced with "4n + k", since the right-hand side of the equation is divided by 4, so that k is any remainder 0 through 3.
How should one approach this problem? What kind of manipulations can you do to floors? What can you assume? etc. etc. ...