🚀 go-pugleaf

For a finite ball packing P in R^3, let 

a(P)=2(number of tangencies)/(number of balls),

then a(P) is the average number of contacts that a ball in P
has.

Conjecture: sup a(P) = 12.

Let A = sup a(P).  Taking large portions of the infinite
cannonball packing shows that A >= 12.  On the other hand,
the following argument shows that A <= 24.  For each contact
c in the packing, let b(c) be the smaller ball from the
two in contact there, or an arbitrary one if the two are equal.
Since in a packing of equal sized balls at most 12 balls can
contact a single ball (I believe this was stated by Newton,
where can a proof be found?), the correspondence
b : contacts -> balls
is at most 12 to 1.  Therefore A<=24.

I think I can improve the naive argument above to get something
better than A<=24, but I don't expect to reach A=12.

Is this problem known?  Any related results?

Oded


oschramm@wisdom.weizmann.ac.il