On Fri, 26 Sep 2003, Dan Hoey wrote:
The way you can guarantee that each ball touches at least two others, one of them previously placed, is this way:
[then what seems to be a correct proof of that statement]
This _allows_ you to search for arrangements by placing each new ball so it touches one previous ball somewhere. If you further restrict your choice. you invalidate the entire effort, since you might not find an arrangement even if one exists.
I looked at trying to modify this for four dimensions, and I find that you can prove that every ball must touch at least two others. Same in K dimensions: every ball must touch at least two others. You might have all N balls in a loop, so you have to guess each ball but the first, second, and last with K-1 degrees of freedom.
Indeed these statements are correct, but (as I can see Dan realises) don't help much in restricting the successive placements of the balls. I don't in fact see how to use them to give a stronger restriction than the simpler one that each ball after the first can be chosen to touch a previous one.
You can't do better with "geometric intuition". John was right, of course.
Thanks for the implied compliment. This discussion should suffice to show why I said that if Musin's argument was of this type, I'd be inclined to disbelieve it. The reason, to make it explicit, is that it's so easy to produce fallacious arguments of this kind that the most probable expectation would be that Musin had done so. Fortunately, it turns out that his argument is of the second kind, and so I expect to find that it's valid. John Conway