Editing Sprouts (game) (section)

===Maximum number of moves===
[[File:Sprouts-max-moves.png|thumb|300px|A game of sprouts with ''n'' initial spots (in blue) that ends in 3''n''&nbsp;−&thinsp;1 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 {{nowrap|3''n'' − ''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 {{nowrap|3''n'' − ''m''}} survivors. There must be at least one survivor, namely the spot added in the final move. So {{nowrap|3''n'' − ''m'' ≥ 1}}; hence a game can last no more than 3''n''&nbsp;−&thinsp;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 3''n''&nbsp;−&thinsp;1 moves.

[[File:Sprouts-analysis.png|right|framed|Live spots (green) and their dead neighbors (black)]]