It is well known that the method of image charges describes how if you have a charge near a conducting plate one can obtain a correct expression for the potential and E field by assuming that (a) the charge is reflected in all nearby conducting surfaces with opposite sign, (b) any image charges in metal plates are themselves reflected, and (c) the E-field always is normal to the conductor.
This works because of the uniqueness and existence theorems of Laplace's equation with the boundary condition of the metal conductor -- and the motion of image charges explains lots of wonderful things, e.g. [Graham]-Smith Purcell radiation can be explained by image charges alone.
My question is this: imagine I have three semi-infinite conducting plates meeting at a vertex 120º apart, with a charge $+Q$ placed on one of the other axes of symmetry.
The image charges I'll get will be placed in the points of an equilateral triangle, and according to the "rules" above they'll be apparently inconsistent: each reflected charge should be both $-Q$ and $+Q$ simultaneously, as can be seen by either going clockwise or anticlockwise around from the solid "real" charge below:
What does the picture of image charges look like, and why? Doesn't it look like the distribution of images is inconsistent with itself? How you reconcile the "classical" image charge explanation that is often given in first-year textbooks with this one. (Likewise, what happens if the angle is anything that would give you an odd number of charges, such as a pentagon rather than a triangle?)
To know what the "right answer" is, I performed some EM simulations by numerically solving Laplace with a charge of +10 C on a 5 mm sphere with d=30 mm, surrounded by either (a) three infinite equipotentials:
This qualitatively is consistent with having two $-Q$ spheres below:
In contrast, the $±Q$ situation seems decidedly wrong (as you'd expect):
As is the $+2Q$ situation:
Why? How do I explain this "simply"?