A thunderbolt in the hammer-nail game: when hammering too hard destroys the support

This paper completes two previous papers on the hammer-nail game. The hammer-nail game goes as follows: two players are in front of a nail slightly driven into a wooden support. Both have a hammer and in turn hit the nail. The winner is the first player able to fully drive the nail into the support. A player is of strength f if he is able, with one swing of the hammer, to drive the nail at most f millimeters into the support. A player is of non dexterity e if he is unable to hammer smoothly, so that, with one swing of the hammer, he drives the nail at least e millimeters into the support, with e >= 1. The two players may be of different strength and dexterity. In the two previous papers we studied this Nim-game by assuming that if the remaining distance is lower than e, then lack of dexterity is not a problem because one swing of the hammer necessarily drives the nail into the support. It followed that strength was more useful than dexterity to win the game. In this paper we suppose that a player destroys the support and loses the game if the remaining distance is lower than e. This new assumption completely changes the results: we now observe that dexterity becomes more useful than strength to win this new hammer-nail game.