Solve the equation $[x / 2]+[x / 4]=x$. ([x], the integer part of $x$ is the greatest integer not exceeding $x$.)
Solution. Let $\frac{x}{4}$ be denoted by $y$, then our equation is: $[2 y]+[y]=4 y$. Since $\left[\frac{x}{2}\right] \leq \frac{x}{2},\left[\frac{x}{4}\right] \leq \frac{x}{4}$, we have $\frac{x}{2}+\frac{x}{4} \geq x$, which means $x \leq 0$, indicating that the equation has no solution among positive numbers.

Let $y=-n+r$, where $n \geq 0$ is the integer part, and $0 \leq r \leq 1$ is the fractional part.

Then $2 y=-2 n+2 r$, so $-2 n+[2 r]-n=-4 n+4 r$. From this,

$$
n+[2 r]=4 r
$$

is an integer, so the possible values of $r$ are: $0, \frac{1}{4}, \frac{1}{2}$, and $\frac{3}{4}$; the possible values of $[2 r]$ are: $0,0,1$, and 1. Thus, from (1), the possible values of $n=4 r-[2 r]$ are: $0,1,1,2$. Therefore, $y=-n+r=0,-\frac{3}{4},-\frac{1}{2}$, or $-\frac{5}{4}$. Thus, the solutions are: $x=4 y=0,-3,-2,-5$.