The player chip counts (i, j, k) can be put in a 3 dimensional space. Due to the constraint i + j + k = N the player chip counts are on a 3D plane. The constraints i >= 0, j >= 0 and k >= 0 further restrict the place of the player chip counts to the equilateral triangle with vertices at (0, 0, N), (0, N, 0) and (0, 0, N). An equilateral triangle is indeed a convenient representation of the 3-players case that displays symmetries.
