Let P be a regular polygon and its interior. Show that for any n > 1, we can find a subset Sn of the plane such that we cannot translate and rotate P to cover Sn but we can translate and rotate P to cover any n points of Sn.
Let the radius of the inscribed circle in P be 1. Let P have m sides. We take Sn to be a circle of radius slightly greater than 1. Clearly it cannot fit inside P. Place it so that its centre is at the centre of P. Suppose it cuts each side a distance tan-1θ either side of the midpoint, where θ is small. Then the arcs lying outside P subtend a total angle 2mθ at the centre. Now rotate the circle about its centre. Let φ measure the angle of rotation from the starting point. In order to keep a given point of the circle inside P, φ must avoid intervals of total length 2mθ. So to keep any given n points of the circle inside P, φ must avoid intervals of total length 2mnθ. The worst case is that these intervals are all disjoint, but provided θ < π/mn they cannot exhaust the available 2π, so we can find angles φ for which all n points are inside P.
17th Putnam 1957
© John Scholes
5 Mar 2002