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Here is a frame that holds circles of radius $1, \frac{1}{2}, \frac13, ..., \frac17$ immobile.

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By "immobile", I mean no circle can move without overlapping other circles or the frame, either individually or simultaneously. (The frame is rigid.)

Can a convex frame hold all circles of radius $1/n$ immobile?

It seems to get increasingly difficult to keep adding circles (and adjusting the frame), while maintaining the conditions that the circles are immobile and the frame is convex. Is there a way to arrange the circles so that you can include all of them?

(This is similar to a question I posted on Math SE. The one on Math SE is potentially much more difficult.)

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  • $\begingroup$ I suspect that for $n$ large one can make a convex frame that holds immobile all the circles in the order $ \ldots, \frac18, \frac16, \frac14, \frac12, 1, \frac13, \frac15, \frac17, \ldots $ (with the $\frac1n$ and $\frac1{n-1}$ circles eventually tangent to each other). There would be plenty of empty space in the interior but no way for any of the circles to reach it without two of them briefly overlapping. $\endgroup$
    – Noam D. Elkies
    Jan 24, 2023 at 5:43

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