  The solitaire of independence is a groupoid action resembling the classical
15-puzzle, which gives information about independent sets of coordinates in a
totally extremally permutive subshift. We study the solitaire with the triangle
shape, which corresponds to the spacetime diagrams of bipermutive cellular
automata with radius 1/2. We give a polynomial time algorithm that puts any
finite subset of the plane in normal form using solitaire moves, and show that
the solitaire orbit of a line of consecutive ones -- the line orbit -- is
completely characterised by the notion of a fill matrix. We show that the
diameter of the line orbit under solitaire moves is cubic.
