1. The problem statement, all variables and given/known data Show that for any real number x, ceiling[(2x + 1) / 2)] - ceiling[(2x + 1) / 4] + floor[(2x + 1) / 4] always equals to floor(x) or ceiling(x). In what circumstances does each case arise? 2. Relevant equations I know that: floor(x) = largest integer ≤ x ceiling(x) = smallest integer ≥ x I also looked at other equations in wikipedia, but don't know where to begin. I tried converting it to another form: ceil(x+0.5) - ceil(x/2+0.25) + floor(x/2+0.25) Then I tried to prove it by cases by assuming that x [itex]\notin[/itex] Z, then x/2+0.25 [itex]\notin[/itex] Z as well. But then I got stuck with ceil(x+0.5) and don't know what to do. ceil(x+0.5) - ceil(x/2+0.25) + floor(x/2+0.25) = ceil(x+0.5) - [2floor(x/2 + 0.25) + 1] Thanks.