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, ${\displaystyle x_{1},\ldots ,x_{N}}$, 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

${\displaystyle x_{1},\ldots ,x_{N}}$

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

${\displaystyle {\frac {k-1}{n}}\leq x_{i}<{\frac {k}{n}}.}$