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

, all between 0 and 1, for which the following conditions hold: Mathematically, we are looking for a sequence of real numbers such that for every n ∈ {1, ..., N} and every k ∈ {1, ..., n} there is some i ∈ {1, ..., k} such that 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: In this example, considering for instance the first 5 numbers, we have Mieczysław Warmus concluded that 768 (1536, counting symmetric solutions separately) distinct sets of intervals satisfy the conditions for N = 17.

A possible solution for N = 17 shown diagrammatically. In each row n , there are n “vines” which are all in different n th s. For example, looking at row 5, it can be seen that 0 < x 1 < 1/5 < x 5 < 2/5 < x 3 < 3/5 < x 4 < 4/5 < x 2 < 1. The numerical values are printed in the article text.