Irregularity of distributions

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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 (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

such that for every n ∈ {1, ..., N} and every k ∈ {1, ..., n} there is some i ∈ {1, ..., n} 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:

A possible solution for N = 17 shown diagrammatically. In each row n, there are n “vines” which are all in different nths. For example, looking at row 5, it can be seen that 0 < x1 < 1/5 < x5 < 2/5 < x3 < 3/5 < x4 < 4/5 < x2 < 1. The numerical values are printed in the article text.

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