page_title: Irregularity of distributions
section_title: Solution
hierarchical_section_title: Irregularity of distributions / Solution
caption_attribution_description: English: Diagram showing one possible solution to the irregularity of distributions problem at N&#160;=&#160;17. There exists no solution for N&#160;≥&#160;18. Deutsch: Darstellung einer möglichen Lösung des Irregularity of distributions-Problems bei N&#160;=&#160;17. Für N&#160;≥&#160;18 gibt es keine Lösung. Français&#160;: Visualisation d’une solution possible du problème d’Irregularity of distributions chez N&#160;=&#160;17. Pour N&#160;≥&#160;18 il n’y a pas de solution. Esperanto: Diagramo montranta unu eblan solucion de la problemo Irregularity of distributions ĉe N&#160;=&#160;17. Ne ekzistas solucio por N&#160;≥&#160;18. Bahasa Melayu: Rajah satu penyelesaian masalah Irregularity of distributions untuk N&#160;=&#160;17. Penyelesaian untuk N&#160;≥&#160;18 tidak wujud. Русский: Одно из возможных решений проблемы Irregularity of distributions для N&#160;=&#160;17. Решений для N&#160;≥&#160;18 не существует.
context_page_description: 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 ∈ {1, ..., N} and every k ∈ {1, ..., n} there is some i ∈ {1, ..., n} such that
context_section_description: 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
caption: 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.
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