Quotients in Ceilings and Floors How would I simplify the expression
$\lceil\frac{2x + 1}{2}\rceil - \lceil\frac{2x + 1}{4}\rceil + \lfloor\frac{2x + 1}{4}\rfloor$
I've tried writing the expression without floors or ceilings, but with no success. I also tried some casework on the parity of x. 

Let me try this way - 
Take $y=\frac{2x+1}4$
If {$y$} $<0.5$, It would reduce to $$(2[y]+1) - ([y]+1) + ([y])   = 0$$
If {$y$} $>0.5$, It would be $$(2[y] +2) - ([y]+1)+([y]) = 1$$
So, now, {$y$}$= 2${$x$}$+0.25$ if {$x$} $<0.375$ 
{$y$}$= 2${$x$}$-0.75$ if $0.875>${$x$} $>0.375$ 
{$y$}$= 2${$x$}$-1.75$ if {$x$} $>0.875$
You will now get the regions of $x$ where this hold good
For $0<${$x$}$<0.125$ The answer is $0$
For $0.125<${$x$}$<0.375$ The answer is $1$
For $0.375<${$x$}$<0.625$ The answer is $0$
For $0.625<${$x$}$<0.875$ The answer is $1$
For $0.875<${$x$}$<1$ The answer is $0$
