I have a loose, perhaps partial explanation for the pattern noticed by Applegate, Jacobson, and Sleator. They used a computer to analyze the game exhaustively through n=11 and noticed that the first player wins if and only if the remainder of n divided by 6 is 0, 1, or 2.
Let me restate this pattern recursively and loosely: there is a tendency for the winner of n = x to be the loser of n = x + 3. Now, in a game of n = x + 3 spots, either of the players is likely to be able to separate 2 untouched spots from a very near equivalent of n = x spots in a single move. (Since this single move will consume 1/3 of an untouched spot.) The winner of n = x spots will likely be the winner of the very near equivalent of n = x spots but will then be the loser of a game consisting of this very near equivalent of n = x spots plus an extra move. Therefore the winner of n = x spots likely will be forced to move to 2 untouched spots so separated. But any move to a formation of 2 untouched spots, or to a formation of 2 untouched spots + 1/3 untouched spot accessible to other spots not accessible to the 2, changes the Sprague-Grundy number of that formation from 0 to 2 or greater, placing the initial mover to that formation at a distinct disadvantage in the total game situation.