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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?

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2 Answers 2

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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 $n=50$: enter image description here

Here's a symmetric-looking solution for $n=48$: enter image description here

Here's a symmetric-looking solution for $n=47$: enter image description here

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    $\begingroup$ What solver did you use? $\endgroup$
    – justhalf
    Dec 17, 2022 at 15:00
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    $\begingroup$ @justhalf I used the NLP solver in SAS. $\endgroup$
    – RobPratt
    Dec 17, 2022 at 20:12
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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.

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  • $\begingroup$ No, the triangle is not equilateral. It's isosceles. Well, you've given me the number of rows to have so that helps. I was also looking for measurement assistance between the lights to place them evenly across the triangle. $\endgroup$
    – Adrien
    Dec 17, 2022 at 2:07

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