This paper first considers questions about games related to Merlin's Magic Square from the point of view of group actions. At this juncture, little beyond the formal model is new, but the exposition sets the stage for considering certain "enhanced" versions of these games. The analysis of enhanced games, with the aid of semigroup actions, is carried out in complete detail for an ostensibly simpler (k = 3) game before turning to a Merlin ( k = 4) game. Concluding sections discuss various ways to generalize our games.
To review the solution to Merlin's Magic Square, we begin by introducing our formal model. As usual, we use 1 to represent an ON light and 0 to represent an OFF light. When light and button are one unit as in Merlin we shall also speak of the button itself as being ON or OFF.
Gary Greenfield. Merlin's Magic Square Enhanced. Technical paper (TR-94-01). Math and Computer Science Technical Report Series. Richmond, Virginia: Department of Mathematics and Computer Science, University of Richmond, January, 1994.