At 04:37 PM 9/25/2003, Jud McCranie wrote:

Use standard backtracking/depth first search.

Plus branch-and-bound to eliminate considering duplicate configurations, e.g. exchanging ball 5 and 8, etc.

On Thu, 25 Sep 2003, Jud McCranie wrote:

At 04:01 PM 9/25/2003, John Conway wrote:

...

Yes, of course. The same argument applies to solving the game of Chess. If next week someone claims to have done that, I won't believe them for the same reason, namely that the problem is so big that their argument is more likely to be wrong than right.

I think a computer search could succeed with the 4-D kissing number problem. I was contemplating it several years ago. Consider for a moment the 3-D version. Take a sphere to be the center sphere. WOLG, place a sphere touching it. WOLG, place the next sphere touching the center sphere and the first one, etc. Use standard backtracking/depth first search. I don't think there are too many possibilities to consider for the 4-D kissing number - far fewer than chess.

This supposes that each sphere touches enough previous ones to effectively define its position, up to symmetry. This supposition does indeed trivialise the proof, but also invalidates it. This perfectly illustrates my point that it's so much easier to produce an invalid proof of this kind than a valid one! This is why if someone comes up with a proof of the case-considering kind, it's more probably going to be wrong than right. In fact there is a continuum of possible positions to be considered for each new sphere after the first few. You might think there'll be a simple argument that reduces this to a finite number, but my guess is that any such argument you produce will be just as wrong as your argument above. Regards, John Conway