18-Point Problem

Place a point somewhere on a Line Segment. Now place a second point and number it 2 so that each of the points is in a different half of the Line Segment. Continue, placing every th point so that all points are on different th of the Line Segment. Formally, for a given , does there exist a sequence of real numbers , , ..., such that for every and every , the inequality

holds for some ? Surprisingly, it is only possible to place 17 points in this manner (Berlekamp and Graham 1970, Warmus 1976).

Steinhaus (1979) gives a 14-point solution (0.06, 0.55, 0.77, 0.39, 0.96, 0.28, 0.64, 0.13, 0.88, 0.48, 0.19, 0.71, 0.35, 0.82), and Warmus (1976) gives the 17-point solution Warmus (1976) states that there are 768 patterns of 17-point solutions (counting reversals as equivalent).

References

Berlekamp, E. R. and Graham, R. L. Irregularities in the Distributions of Finite Sequences.'' J. Number Th. 2, 152-161, 1970.

Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, pp. 34-36, 1997.

Steinhaus, H. Distribution on Numbers'' and Generalization.'' Problems 6 and 7 in One Hundred Problems in Elementary Mathematics. New York: Dover, pp. 12-13, 1979.

Warmus, M. A Supplementary Note on the Irregularities of Distributions.'' J. Number Th. 8, 260-263, 1976.