'Twas the night before Christmas and throughout the net,
    Not a question was posed, at least---not yet.
    When what to my horror I suddenly realized:
    One present was not wrapped, a present most prized.
    For a juggler of the future, three balls all the same.
    But how best to foil-wrap, within a tight frame?


> ***Q1***. What is the smallest square that can wrap three unit-radius balls?

To *wrap* means to completely cover their convex hull.
I mention "foil" above because one may want to crinkle the
wrapping over sphere caps, analogous to how
[Mozartkugeln](https://en.wikipedia.org/wiki/Mozartkugel) are wrapped.<sup>1</sup>
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&nbsp; &nbsp; &nbsp;
[![Balls3][1]][1]
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Likely easier is this question, which may only require rough bounds:

> ***Q2***. Which of the two configurations shown above is easier to wrap,
easier in the sense that a smaller square suffices?



<hr />
<sup>1</sup>
The square of diagonal $2 \pi$ is the smallest square that wraps a unit-radius sphere.
Demaine, Erik D., Martin L. Demaine, John Iacono, and Stefan Langerman. 
"Wrapping spheres with flat paper." *Computational Geometry* 42, no. 8 (2009): 748-757.
[Journal link](https://www.sciencedirect.com/science/article/pii/S0925772109000182).


  [1]: https://i.sstatic.net/dGfCz.jpg