of the number of spots. Then, it can be shown that if the game starts with n spots, it will end in no more than 3n−1 moves and no fewer than 2n moves. In the following proofs, it is assumed that a game starts with n spots and lasts for exactly m moves. Maximum number of moves Each spot starts with three lives and each move reduces the total number of lives in the game by one (two lives are lost at the ends of the line, but the new spot has one life). So at the end of the game there are 3n−m remaining lives. Each surviving spot has only one life (otherwise there would be another move joining that spot to itself), so there are exactly 3n−m survivors. There must be at least one survivor, namely the spot added in the final move. So 3n−m ≥ 1; hence a game can last no more than 3n−1 moves. This upper bound is actually the maximum, and it can be attained in many ways by ensuring that there is only one survivor at the end of the game. For instance, the game on the right has one survivor and 3n−1 moves. Minimum number of moves At the end of the game, a dead spot is called the neighbor of a survivor if it is either adjacent to that survivor or, if the survivor has a loop, it is adjacent to a spot adjacent to the survivor. This is illustrated in the diagram to the right. Each survivor has exactly two dead neighbors. No dead spot can be the neighbor of two different survivors, for otherwise there would be a move joining the survivors. All other dead spots (not neighbors of a survivor) are called pharisees (from the Hebrew for "separated ones"). Suppose there are p pharisees. Then since initial spots + moves = total spots at end of game = survivors + neighbors + pharisees. Rearranging gives: Consequently, a game lasts for at least 2n moves, and the number of pharisees is divisible by 4. This lower bound on the length of a game is actually the minimum. The diagram on the right shows a completed game of 2n moves. It has n survivors, 2n neighbors and 0 pharisees. Importance in real games Real games seem to turn into a battle over whether the number of moves will be k