As the title suggests, the general solution of Laplace's equation has 4 arbitrary constants. One would imagine that if you e.g. have the potential at 4 points in a domain, you could get the specific solution by replacing: V(x1,y1)=V1, V(x2,y2)=V2, V(x3,y3)=V3, V(x4,y4)=V4, and solving the 4x4 system to find the 4 arbitrary constants. I tried it but it doesn't yield a solution. Intuitively I understand that since I have not provided equations for the boundaries I should not be able to get the solution, but does anyone know the actual mathematical cause (I suspect that the equations are linearly dependent, but no sure why)?