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,, all between 0 and 1, for which the following conditions hold:
  • The first two numbers must be in different halves.
  • The first 3 numbers must be in different thirds.
  • 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
such that for every n ∈ and every k ∈ there is some i ∈ such that

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:
In this example, considering for instance the first 5 numbers, we have
Mieczysław Warmus concluded that 768 distinct sets of intervals satisfy the conditions for N = 17.