# Points inside a triangle of guaranteed distance

My question starts from a problem in Arthur Engel's problem solving strategies, Chapter 4 : Box principle,

A target has the form of an equilateral triangle with side 1.

(a) If it is hit 5 times, then there will be two holes with distance ≤ 1/2.

(b) It is hit 17 times. What is the minimal distance of two holes at most?

So, these can be solved by pigeon hole principle application. Now I thought of some question based on these two questions, and I need your help with it.

Given such a triangle as stated above, if we hit it m times, then there will be two holes with distance ≤ d. And d is the least such possible distance, among all possible distances. Find the relation between m and d.

So, for the cases when, m = $4^n+1$ , d = $\frac{1}{2^n}$ $\forall n \in \mathbb{N}$.

But, for the other cases I have no idea as to how to go about solving it! To me it seems clear that I need to be able to come up with some partitioning of the triangle area such that I can allow for the similar pigeon hole argument. Unfortunately, I am unable to execute this intuitive idea.

It'd really helpful if anyone could guide me in some direction as to how I should go about it!

• when $m=5$, $d=1$ also works, so what you really want is relationship between $m$ and least such $d$ Commented Jul 21, 2017 at 17:42