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 Nth point so that all N points are on different (1/N)th of the line segment. Formally, for a given N, does there exist a sequence of real numbers x_1, x_2, ..., x_N such that for every n in {1,...,N} and every k in {1,...,n}, the inequality


holds for some i in {1,...,n}? 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).

See also

Discrepancy Theorem, Point Picking

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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.

Referenced on Wolfram|Alpha

18-Point Problem

Cite this as:

Weisstein, Eric W. "18-Point Problem." From MathWorld--A Wolfram Web Resource.

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