Irregularity of distributions

The irregularity of distributions problem, stated first by Hugo Steinhaus, is a numerical problem with a surprising result. The problem is to find N numbers, $$x_1,\ldots,x_N$$, all between 0 and 1, for which the following conditions hold:


 * The first two numbers must be in different halves (one less than 1/2, one greater than 1/2).
 * The first 3 numbers must be in different thirds (one less than 1/3, one between 1/3 and 2/3, one greater than 2/3).
 * The first 4 numbers must be in different fourths.
 * The first 5 numbers must be in different fifths.
 * etc.

Mathematically, we are looking for a sequence of real numbers


 * $$x_1,\ldots,x_N$$

such that for every n &isin; {1, ..., N} and every k &isin; {1, ..., n} there is some i &isin; {1, ..., k} such that


 * $$\frac{k-1}{n} \leq x_i < \frac{k}{n}.$$

Solution
The surprising result is that there is a solution up to N = 17, but starting at N = 18 and above it is impossible. A possible solution for N ≤ 17 is shown diagrammatically on the right; numerically it is as follows:





\begin{align} x_{1} & = 0.029 \\ x_{2} & = 0.971 \\ x_{3} & = 0.423 \\ x_{4} & = 0.71 \\ x_{5} & = 0.27 \\ x_{6} & = 0.542 \\ x_{7} & = 0.852 \\ x_{8} & = 0.172 \\ x_{9} & = 0.62 \\ x_{10} & = 0.355 \\ x_{11} & = 0.777 \\ x_{12} & = 0.1 \\ x_{13} & = 0.485 \\ x_{14} & = 0.905 \\ x_{15} & = 0.218 \\ x_{16} & = 0.667 \\ x_{17} & = 0.324 \end{align} $$

In this example, considering for instance the first 5 numbers, we have


 * $$0 < x_1 < \frac{1}{5} < x_5 < \frac{2}{5} < x_3 < \frac{3}{5} < x_4 < \frac{4}{5} < x_2 < 1.$$

Mieczysław Warmus concluded that 768 (1536, counting symmetric solutions separately) distinct sets of intervals satisfy the conditions for N = 17.