If $x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor=2018$, find $x$ If $x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor=2018$, find $x$.
My working: if $x$ is positive then by estimation it must be in $(6,7)$ and for this interval I : $x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor<2003$. So no solution for positive $x$.
For negative $x$, I got $x \in (-7,-6)$ and using this I got $x=\frac{2018}{k}, k$ can be $-336,-335,\cdots,-289,$
Now using excel sheet I got $x=-\frac{2018}{305}$.
 A: $$x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor=2018$$
Let $f(x) = x\Big\lfloor x \big\lfloor x\lfloor x \rfloor \big\rfloor \Big\rfloor$
Let $\xi = -\dfrac{6862735}{1037232} = -6 \dfrac{639343}{1037232}$
Then $f(\xi) - 2018 = -\dfrac{1}{1037232} = -\dfrac{1}{2^4 3^3 7^4}$
Let $x = \xi + \delta$. I want to solve $f(x)-2018 = 0$ for $\delta$.
I will assume that $\delta$ is very small. What I mean by that will make itself clear as I progress.
$\lfloor x \rfloor = -7$
$x \lfloor x \rfloor 
   = -7\xi - 7\delta  
   = 46 \dfrac{46639}{148176} - 7\delta$
$\lfloor x \lfloor x \rfloor \rfloor = 46$
$x \lfloor x \lfloor x \rfloor \rfloor 
   = 46\xi + 46 \delta
   = -304 \dfrac{183641}{518616} + 46 \delta$
$\lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor = -305$
$x \lfloor x \lfloor x \lfloor x \rfloor \rfloor \rfloor 
   = -305\xi -305\delta
   = 2017 \dfrac{1037231}{1037232} - 305 \delta $
\begin{align}
   f(x) &= 0 \\
   \left(2017 \dfrac{1037231}{1037232} - 305 \delta \right) - 2018 &= 0 \\
   305 \delta &= -1/1037232 \\
   \delta &= -\dfrac{1}{316355760}
\end{align}
$\color{red}{x = \xi + \delta = -\dfrac{2018}{305} = -6 \dfrac{188}{305}}$
