I can satisfy myself that 4 is too few, and 6 is enough. I suspect 5 kissing spheres always admit a 6th, but I haven't been able to prove it. So "5 or 6" is as close as I can get to an answer at the moment. On Mon, Jun 1, 2020 at 11:02 PM Dan Asimov <dasimov@earthlink.net> wrote:

Aha. My copy (3rd edition) is not at hand. Can you please mention the page and/or chapter where that's found?

—Dan

----- Conway's Sphere Packing, Lattices and Groups has an interesting (and surprising!) discussion of this particular case. -----

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