Possible locations of the treasure chest

Question

You are at location $(0,0)$ and know the treasure is within $100$m of you.

Person A said that from $(0,0)$ they walked $108$m to where the treasure was but their distance is calculated by $|x|$ + $|y|$.

Person B said that from $(0,0)$ they walked $105$m to where the treasure was but their distance is calculated by $\frac{|x|}{\sqrt{3}}$ + $\max\left\{\frac{|x|}{\sqrt{3}}, |y|\right\}$.

How many locations are their for the treasure?

I was thinking you write these as simultaneous equations and solve them from there. But the taking bigger value kind of messes up the equations.

Answer 1

Hint: Things are symmetric for positive and negative $x$ and $y$, so just assume $x\ge 0$ and $y \ge 0$ and solve the problem. Then each solution where $x$ and $y$ are nonzero gives four solutions, each solution where exactly one is zero gives two, and the origin (if it were a solution, which it is not here) gives one. For the last max, just do the cases. Assume $\frac x{\sqrt 3} \gt y$, solve, and check that the solution meets the assumption, then assume the other way.

Answer 2

Your first equation translates to $\sqrt{x^2 + y^2} = 100 \iff x \in \{\sqrt{10^4 - y^2},-\sqrt{10^4 - y^2}\} $

Then $|x|+|y|=\left|\pm \sqrt{10^4 - y^2}\right|+|y|=\sqrt{10^4 - y^2}+|y|=108$

A simple calculation shows that there are only 4 possible values of $y$ that satisfy $\sqrt{10^4 - y^2}+|y|=108$. For each case we can compute at most 2 possible values for $x$ using $|x|+|y|=108$. Then we can just check the inequality with the maximum.