Circle the wagons

Geometry Level 5

Given a circle with a diameter of 100.

Twelve points $P_1, \dots, P_{12}$ lies on a circle, in such a way that their total pairwise distance, $D = \sum_{i < j} d(P_i, P_j)$ is maximal.

Give the value of $\lfloor D \rfloor$ . (I.e. rounded down to the nearest integer.)

The answer is 4557.

1 solution

Arjen Vreugdenhil
Dec 8, 2015

The distance is optimized when the points lie evenly distributed around the circle, i.e. at $30^\circ$ intervals.

The distance between two points on a circle of radius $R$ at distance $\theta^\circ$ is $d = 2R\sin\frac{\theta}2.$

There are 12 pairs of points that are $30^\circ$ apart. Their distance is $d_1 = 100 \sin 15^\circ = 25.8819.$ Likewise, there are 12 pairs at $60^\circ$ ; 12 pairs at $90^\circ$ ; 12 pairs at $120^\circ$ ; and 12 pairs at $150^\circ$ : $d_2 = 100 \sin 30^\circ = 50.0000, \\ d_3 = 100 \sin 45^\circ = 70.7107, \\ d_4 = 100 \sin 60^\circ = 86.6025, \\ d_5 = 100 \sin 75^\circ = 96.5926.$ Finally, there are 6 pairs of point diametrically opposite, with distance $d_6 = 100.0000$ . The total is $D = 12\cdot (25.8819 + 50.0000 + 70.7107 + 86.6025 + 96.5926) + 6\cdot 100.0000 = \boxed{4557}.452.$

Given that the total distance is maximized with the twelve points at the vertices of a dodecagon, the calculation is, of course, easy. The nontriviality of this question lies in showing that this configuration is the one that maximizes the total distance. You just state the result.

It was proved for $n$ points by Toth, "On the sum of distances determined by a pointset", Acta Math. Acad. Sci. Hungar., 7 (1956), pp. 397–401.

Mark Hennings - 5 years, 6 months ago

Circle the wagons

The answer is 4557.

1 solution

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