[Retros] Not a Re:tro puzzle

[Noam Elkies]

Andrew writes:


> Suppose that W & B (the kings) are distinct points in the plane,

> and we can choose the location of other distinct points w_1,...,w_k,

> (white points) & b_1,...,b_l (black points).  [...]


Richard Stanley replied:


> Perhaps Andy meant that not all the points lie on a straight line.


[to avois linear counterexamples such as Mark Tilford's]


> In this case the convex hull of the points (the smallest convex

> polygon containing all the points, either on the boundary or inside)

> has at least three vertices.  One of these vertices cannot be a king.

> This vertex is not pinned, so there is always at least one unpinned point.


Probably easier to see this by considering a point at maximal distance from
the line joining the kings.  The maximum exists because the set of points
is nonempty and finite; and as long as this maximal distance is positive,
such a point is not a king but cannot be pinned even if the maximum is
shared by another point in the set.

NDE