[Retros] a chess-related math puzzle
Richard Stanley
rstan at math.mit.edu
Sun Mar 23 23:33:43 EDT 2008
> 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).
>
> Say a white point w_i is *pinned* if there exists j
> such that w_i lies on the line segment between b_j &
> W. Similarly define pinning for the black points b_j.
>
> Can there exist a non-empty set of white & black
> points all of which are pinned? If yes show one, if
> not prove it.
Perhaps Andy meant that not all the points lie on a straight line. 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.
Richard
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