Find all real numbers $x$ such that $x[x[x[x]]]=88$ where $[\cdot]$ denotes floor function. 
Question: Find the all real numbers $x$ such that $x[x[x[x]]]=88$ where $[\cdot ]$ denotes floor function.

Attempt: $$x[x[x[x]]]=88\implies [x[x[x]]]=\frac{88}{x}.$$
Since left side of equation always gives an integer value, $88$ has to be divisible by $x$. 
Also, $3^4=81$. So the solution should lie between $3$ and $4$. 

So, the given equation basically implies $x^4\approx88$, and we know $\frac{88}x$ is an integer. Now, if $x^4\approx88$, then $x\approx\sqrt[4]{88}=3.063$, and then $\frac{88}x\approx28.732$. So I'd guess either $\frac{88}x=28$ or $\frac{88}x=29$. That is, I'd guess either $x=\frac{88}{28}=\frac{22}7$ or $x=\frac{88}{29}$. No guarantees that one of them works, though.
We do know there's at most one solution, since $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor$ is an increasing function.
