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

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

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

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:

A possible solution for N = 17 shown diagrammatically. In each row n, there are n “vines” which are all in different n^ths. 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.

{\begin{aligned}x_{1}&=0.029\\x_{2}&=0.971\\x_{3}&=0.423\\x_{4}&=0.71\\x_{5}&=0.27\\x_{6}&=0.542\\x_{7}&=0.852\\x_{8}&=0.172\\x_{9}&=0.62\\x_{10}&=0.355\\x_{11}&=0.777\\x_{12}&=0.1\\x_{13}&=0.485\\x_{14}&=0.905\\x_{15}&=0.218\\x_{16}&=0.667\\x_{17}&=0.324\end{aligned}}

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

0<x_{1}<{\frac {1}{5}}<x_{5}<{\frac {2}{5}}<x_{3}<{\frac {3}{5}}<x_{4}<{\frac {4}{5}}<x_{2}<1.

Mieczysław Warmus concluded that 768 (1536, counting symmetric solutions separately) distinct sets of intervals satisfy the conditions for N = 17.

References

H. Steinhaus, One hundred problems in elementary mathematics, Basic Books, New York, 1964, page 12
Berlekamp, E. R.; Graham, R. L. (1970). "Irregularities in the distributions of finite sequences". Journal of Number Theory. 2 (2): 152–161. Bibcode:1970JNT.....2..152B. doi:10.1016/0022-314X(70)90015-6. MR 0269605.
M. Warmus, "A Supplementary Note on the Irregularities of Distributions", Journal of Number Theory 8, 260–263, 1976.

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