Oh! You count countries that are only adjacent at corners as neighbours? That changes things a lot. (And surely isn’t the usual definition.) I think you should make that explicit in the statement of the problem.
It doesn’t count in the discussions of coloring graphs, such as in the four color map theorem, and that’s the kind of math this is most similar to. So you really need to specify.
My immediate reaction, without actually doing any calculation or diagram-drawing or whatever, is:
Jnvg, jung?, qbrfa’g gur nirentr ahzore bs arvtuobhef unir gb or fgevpgyl yrff guna 6 sbe ernfbaf qrevivat sebz Rhyre’f sbezhyn? Vs fb, gur nafjre vf gung gur fvghngvba qrfpevorq vf vzcbffvoyr, ertneqyrff bs ubj znal ynlref bs ynaqybpx lbh unir.
But the fuzzy combination of intuition and hazy memory that tells me that may be all wrong.
Well, imagine the chessboard. There are 36 fields with 8 neighbours, 24 with 5 neighbours, and 4 with 3 neighbours.
Which gives you an average greater than 6.
There is a quadripoint between Botswana, Namibia, Zambia and Zimbabwe, for example.
Oh! You count countries that are only adjacent at corners as neighbours? That changes things a lot. (And surely isn’t the usual definition.) I think you should make that explicit in the statement of the problem.
Perhaps, but it follows.
It happens also at the Game of Life by Conway, at the game of chess and at many related games, as well as in real life geography.
It doesn’t count in the discussions of coloring graphs, such as in the four color map theorem, and that’s the kind of math this is most similar to. So you really need to specify.
Okay. The next time I’ll be more careful to eliminate any possible ambiguity in advance.