In his book of problems from elementary mathematics Hugo Steinhaus asked the following:
Is it possible to find an infinite sequence of real numbers $x_1,x_2,...$, such that $x_1$ lies in the interval [0,1); each of $x_1,x_2$ lies in a different interval from among [0,$\frac{1}{2}$), [$\frac{1}{2}$,1); each of $x_1,x_2,x_3$ lies in a different interval from among [0,$\frac{1}{3}$), [$\frac{1}{3}$,$\frac{2}{3}$),[$\frac{2}{3}$,1), ...? Or if not how long can such a sequence be?
Another way to ask the same question is:
Does there exist for every positive integer N 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} we have
$\frac{k-1}{n}$ <= $x_i$ <$\frac{k}{n} $ for some i in {1,...,n}? If not, what is the greatest possible N?
Using a computer this is easy enough to do. The answer is that no more than 17 such points can be found. See M. Warmus, A Supplementary Note on the Irregularities of Distributions. Journal of Number Theory 8, 260-263 (1976).
I've tried to extend this problem in several different ways, but so far the searches involved have been too great for my computer. I think that there is a chance to compute the following generalization.
How long a sequence $x_1,x_2,...,x_N$ can one find such that $x_1$ lies in the interval [0,1); each of $x_1,x_2,x_3$ lies in a different interval from among [0,$\frac{1}{3}$), [$\frac{1}{3}$,$\frac{2}{3}$),[$\frac{2}{3}$,1); each of $x_1,x_2,x_3,x_4,x_5$ lies in a different interval from among [0,$\frac{1}{5}$), [$\frac{1}{5}$,$\frac{2}{5}$),[$\frac{2}{5}$,$\frac{3}{5}$),[$\frac{3}{5}$,$\frac{4}{5}$),[$\frac{4}{5}$,1),...?
