K-color region select game

The region select game, introduced by Ayaka Shimizu, Akio Kawauchi and Kengo Kishimoto, is a game that is played on knot diagrams whose crossings are endowed with two colors. The game is based on the region crossing change moves that induce an unknotting operation on knot diagrams. We generalize the region select game to be played on a knot diagram endowed with $k$-colors at its vertices for $2 \leq k \leq \infty$.