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. 

If $2x+1=4t+u,0<u<4$  where $t$ is any integer
$\lceil\frac{2x + 1}2\rceil - \lceil\frac{2x + 1}4\rceil + \lfloor\frac{2x + 1}4\rfloor=\lceil\dfrac u2\rceil-\lceil\dfrac u4\rceil-\lfloor\dfrac u4\rfloor$
Now $0<u<4\iff0<\dfrac u2<2$
For $0<\dfrac u2\le1,\lceil\dfrac u2\rceil=1$ and for 
$1<\dfrac u2<2,\lceil\dfrac u2\rceil=2$
$0<u<4\iff0<\dfrac u4<1\implies\lceil\dfrac u4\rceil=1$ and $\lfloor\dfrac u4\rfloor=0$
