I am trying to find the conditions under which $\lfloor(n-1)x\rfloor + \lfloor x \rfloor = \lfloor nx \rfloor$. The trivial case is whenever $x \in \mathbb{Z}$. If $n = 2$, then $x - \lfloor x \rfloor \lt \frac{1}{2}$.
As $n$ increases however, the pattern becomes more complicated. For example, for $n = 3$, if $x - \lfloor x \rfloor \lt \frac{1}{3}$ then $\lfloor 3x \rfloor = 3\lfloor x \rfloor$, and $\lfloor 2x \rfloor = 2\lfloor x \rfloor$ so the identity is satisfied. However, if $ \frac{1}{2} \lt x - \lfloor x \rfloor \lt \frac{2}{3}$, then the identity is also satisfied.
I tried seeing whether Hermite's identity can help but I can't see an obvious way of applying it.