World
Game
Of
Sprouts
Association
A cannibal is a spot with zero or two lines attached, while a ghost is a spot with one line attached. Any particular move increases or decreases the total number of cannibals by an odd number. A move from a cannibal to a cannibal increases or decreases the total number of cannibals by one. A move from a ghost to a ghost increases the total number of cannibals by one or three. A move from a cannibal to a ghost increases the total number of cannibals by one.
A pair of successive moves will increase or decrease the total number of cannibals by an even number, since an odd number plus an odd number is an even number. Accordingly, a particular player in a particular game will always be faced with an even number of cannibals or an odd number of cannibals. It would be impossible for a player to receive a position on one move in which there are an even number of cannibals and receive a position on another move in which there are an odd number of cannibals. Perhaps we should call this phenomenon the “parity conservation principle”. All of the positions which a given player “fields” will be of the same parity in regards to number of cannibals.
The parity conservation principle applies not only to whole positions, but also to subsets of whole positions. Say there is a trap or a potential trap on the board. A trap is a biosphere that will produce the second player’s choice of an even or odd number of survivors. We normally avoid moving to a trap, because we do not want to give our opponent the opportunity to make such a choice. An important question is, then, who is going to make the last move outside of the trap? Just as in the position as a whole, it all comes down to survivors. Survivors are cannibals, so a position or a subset B of a position which ends up with S survivors and no other live spots will never be fielded by a given player, if that player at any time has a position or subset B of a position containing C cannibals such that S and C are of different parity.
If there is a trap in a position, I say that there is a “cannibal war” centered on that trap. Each player will normally try to get the nontrap part of the position to produce a favorable number of survivors.