<s>[INST] Calculation of $x$ in $x \lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = 88$ How can I calculate real values of $x$ in $x \lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = 88$, where $\lfloor x\rfloor$ is the floor function?
My attempt:
Let $\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = k\in \mathbb{Z}$. Then
$$
k \leq x\lfloor x\lfloor x\rfloor\rfloor<k+1
$$
and our equation becomes
$$
x\cdot k = 88 \implies x=\frac{88}{k}
$$
For $x>0$, simple guessing shows that $3.1<x<3.2$. But how can we account for $x<0$?
 [/INST] Through guess and check 
$x=\dfrac{22}{7}$
We know apriori, $x\approx 3.16 \Longrightarrow \lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = 28$, so setting $k=28$ yields $x=22/7$
You can verify this yourself:
$$
\tfrac{22}{7}\lfloor\tfrac{22}{7}\lfloor \tfrac{22}{7}\lfloor \tfrac{22}{7}\rfloor\rfloor\rfloor =88 
$$
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