For what values of $n$ can circular coins of radius $\frac12,\frac13,\frac14,...,\frac1n$ (at least one of each, and no other kind of coin) be held rigidly in a circular tray of radius $1$?
By "held rigidly", I mean that no coin can move, except that each coin can rotate about its own centre, and all the coins can rotate collectively about the tray's centre.
My attempt
I know that $n$ can equal $2,3,4$:
For $n>4$, I suspect there are no solutions, but I cannot prove it.
For $n=5$, the "closest" I have gotten is the following (here is the desmos graph).
Below is a perfect fit with $\frac12,\frac13,\frac14\,\frac16,\frac17$. Missing $\frac15$, so no good.
Context:
This is the latest question of mine related to rigidly held coins.
The original question was: "Five circles in a rectangle: Can the circles move?"
The previous question was on Puzzling Stack Exchange: "In a circular tray of radius 1, arrange coins of radius $\frac12,\frac13,\frac14,\frac15$ so that none of them can move independently" (Note: The Puzzling question uses a different kind of rigidity than my question here.)
