Editing Sprouts (game) (section)

===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 language|Hebrew]] for "[[Pharisees|separated ones]]"). Suppose there are ''p'' pharisees. Then

:<math>n + m = 3n - m + 2(3n - m) + p</math>

since initial spots + moves = total spots at end of game = survivors + neighbors + pharisees. Rearranging gives:

:<math> m = 2n + p/4</math>

Consequently, a game lasts for at least 2''n'' moves, and the number of pharisees is [[divisible]] by 4.

[[File:Game of sprouts with n initial vertices, ending in minimum number of moves.png|thumb|300px|A game of sprouts with ''n'' initial spots that ends in 2''n'' moves]]
This lower bound on the length of a game is actually the minimum. The diagram on the right shows a completed game of 2''n'' moves. It has ''n'' survivors, 2''n'' neighbors and 0 pharisees.