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

Question

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 $4$.

$x$ doesn't have to be an integer just because $88/x$ is, but it does greatly decrease the number of possible options. There are only finitely many between $3$ and $4$. — Arthur, Nov 10 '15 at 11:22
There is a solution, but it's not an integer. $\frac{88}x$ is an integer, though. — Akiva Weinberger, Nov 10 '15 at 11:22
Possible duplicate of Calculation of $x$ in $x \lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor = 88$ — Jyrki Lahtonen, Sep 27 '19 at 03:59

score 9 · Accepted Answer · answered Nov 10 '15 at 11:32

9

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.

answered Nov 10 '15 at 11:32

Akiva Weinberger

23,062

88/27 is the answer given in my book. – Aditya Dev Nov 12 '15 at 05:02
1

@AdityaDev Really? Wolfram Alpha says otherwise. I'm pretty sure that $88/28=22/7$ should be the answer; perhaps it was a typo? – Akiva Weinberger Nov 12 '15 at 05:24
Typo.... It's 88/28 – Aditya Dev Nov 12 '15 at 05:47

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

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