# [Retros] Not a Re:tro puzzle

Noam Elkies elkies at math.harvard.edu
Sun Mar 23 23:40:58 EDT 2008

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