Possible locations of the treasure chest 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.

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.