How to geometrically determine number of holes for a triangle

Question

all.

I am making a modern Christmas tree out of a flat wooden board. It's triangular in shape. It's real simple and modern. It measures 44" from top to bottom and at the widest point it's also 44". I have a strand of 50 LED bulbs that I want to geometrically place by drilling holes through the wood to poke them trough (so only the bulb is seen on the front surface. Is there some type of equation I can use to equally place fifty holes throughout my triangle? I want to start with a single hole near the top point and then gradually increase them with equal distance as I work my way down.

Worse case scenario, if I have to have a couple less holes than 50 I could leave the remaining bulbs behind the triangle tree. No one will see what's behind it. But I cannot go over 50 since I only have 50 lights.

Can I get some assistance, please?

Answer 1

One approach is to maximize the dispersion (the minimum distance between any pair of holes). Explicitly, this (nonconvex) nonlinear optimization problem is to maximize $z$ subject to \begin{align} z &\le (x_i-x_j)^2 + (y_i-y_j)^2 &&\text{for $1\le i<j\le 50$} \\ y_i &\le 44 - 2x_i &&\text{for $i\in[50]$} \\ y_i &\le 44 + 2x_i &&\text{for $i\in[50]$} \\ y_i &\ge 0 &&\text{for $i\in[50]$} \end{align}

Here's a solution for $n=50$:

Here's a symmetric-looking solution for $n=48$:

Here's a symmetric-looking solution for $n=47$:

Answer 2

The largest triangular number that does not exceed $50$ is $45$, meaning that you can have nine rows of lights, in which the first row has one light, the second row has two, the third row has three, and so forth, until the last row on the bottom has nine lights.

However, if your triangular wooden board is not equilateral (it doesn't sound like it is), then the spacing between lights in adjacent rows will not be the same as the spacing between adjacent lights in the same row. Whether or not this is acceptable to you depends on your personal taste.