A singular quartic curve over a finite field and the Trisentis game

The Trisentis game consists of a rectangular array of lights each of which also functions as a toggle switch for its (up to eight) neighboring lights. The lights are OFF at the start, and the object is to turn them all ON. We give explicit formulas for the dimension of the kernel of the Laplacian associated to this game as well as some variants, in some cases, by counting rational points of the singular quartic curve (x+x−1+1)(y+y−1+1)=1 over finite fields. As a corollary, we have an affirmative answer to a question of Clausing whether the n×n Trisentis game has a unique solution if n=2⋅k4 or n=2⋅k4−2.