The Sprouts winning-strategy congruence conjecture

In the Sprouts game, a two-player game beginning with $p$ spots in which the player making the last move wins, a player has a winning strategy for exactly one of the two players. Sprouts winning-strategy conjecture. The first player has a winning strategy if and only if $p$ is congruent to $3$, $4$, or $5$ modulo $6$. This is the original pattern conjectured from computer analysis of the game; the paper studies its extension from the plane to compact surfaces, while the general status of the pattern is not resolved here.