From 3046ff170a03818c3220535d8a339e091e83c33f Mon Sep 17 00:00:00 2001
From: Vectornaut <vectornaut@nobody@nowhere.net>
Date: Fri, 23 May 2025 21:43:35 +0000
Subject: [PATCH] Start porting test problems from the "User Stories" page

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+Here are some objects one might explore and problems one might solve with dyna3. We'll use this list to guide and evaluate our progress in developing the dyna3 engine and interface. We're striving to build an engine that can represent and solve as many of these problems as possible, and an interface that makes them as convenient as possible to set up.
+
+## Basic elements and constraints
+
+### Rigid Hexagon
+
+#### Source
+
+- **Author:** Noam Elkies
+- **Published:** P12 from [the Playground in *Math Horizons*: Volume 2, Issue 1](https://www.tandfonline.com/doi/abs/10.1080/10724117.1994.11974897)
+
+#### Statement
+
+“(Proposed by Noam Elkies, Harvard University.) Suppose $A$, $B$, $C$, $D$, $E$, $F$ are points in 3-space such that
+```math
+\begin{alignat*}{6}
+AB &\;=\;& CD &\;=\;& EF &\;=\;& r, \\
+BC &\;=\;& DE &\;=\;& FA &\;=\;& s, \\
+AD &\;=\;& BE &\;=\;& CF &\;=\;& r + s
+\end{alignat*}
+```
+for some positive $r$, $s$. Show that $A$, $B$, $C$, $D$, $E$, $F$ must be the vertices of an equiangular planar hexagon.”
+
+## Hierarchical constraints
+
+These problems impose various kinds of *soft constraints* on top of the *hard constraints* that an assembly must satisfy to qualify as a solution. Here are some possible kinds of soft constraints.
+
+- **Optimizing constraints:** Loss function terms whose sum should be minimized.
+- **Optional constraints:** Constraints that may be relaxed if they can't be satisfied.
+
+### Frugal Firepower (optimizing)
+
+#### Source
+
+- **Author:** David Seppala-Holtzman
+- **Published:** P370 from [the Playground in *Math Horizons*: Volume 25, Issue 4](https://digitaleditions.sheridan.com/article/The+Playground/3039776/483313/article.html)
+
+#### Statement
+
+“David Seppala-Holtzman of St. Joseph’s College New York gives us the following problem. A customer orders five identical perfectly spherical cannonballs from Adderley’s Cannonball Emporium, and it’s your job to pack them for shipping. The Emporium ships only in rectangular boxes but can construct such boxes with any desired dimensions. You have a choice of packing the cannonballs so that their centers form a square pyramid or two triangular pyramids, as in figure 1 [see source], but you can orient the arrangements however you like inside the box.  The shipping cost will be proportional to the sum of the length, width, and height of the box.  In **Frugal Firepower**, determine which arrangement allows you to minimize the shipping cost.”
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