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Problem of the Week 854

A Circular Balancing Act

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If P is a point on a circle, then P' refers to the diametrically opposite point.
  1. Arrange the integers from 1 to 1998 uniformly around a circle so that a+b = a'+b' whenever a and b are adjacent.
  2. How many such "balanced" arrangements of 1, 2, ..., 1998 are there?
Extra credit for enthusiasts: Solve the problem with an arbitrary even integer replacing 1998.

Source: N. J. Finizio and J. T. Lewis, College Math. Journal (25) (Jan 1994) Problem 492, p. 66.

© Copyright 1998 Stan Wagon. Reproduced with permission.

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2 October 1998