174.30.131.200: /* Equation in the extended complex plane */

2012-04-24T06:51:51Z

Equation in the extended complex plane

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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, <math>x_1,\ldots,x_N</math>, 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 number]]s

:<math>x_1,\ldots,x_N</math>

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

:<math>\frac{k-1}{n} \leq x_i < \frac{k}{n}.</math>

== 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:

[[Image:Irregularity of distributions.png|thumb|right|400px|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 < ''x''1 < 1/5 < ''x''5 < 2/5 < ''x''3 < 3/5 < ''x''4 < 4/5 < ''x''2 < 1. The numerical values are printed in the article text.]]

: <math>
\begin{align}
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} & = 0.774 \\
x_{12} & = 0.114 \\
x_{13} & = 0.485 \\
x_{14} & = 0.926 \\
x_{15} & = 0.207 \\
x_{16} & = 0.677 \\
x_{17} & = 0.297
\end{align}
</math>

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

: <math>0 < x_1 < \frac{1}{5} < x_5 < \frac{2}{5} < x_3 < \frac{3}{5} < x_4 < \frac{4}{5} < x_2 < 1.</math>

==References==
* H. Steinhaus, ''One hundred problems in elementary mathematics'', [[Basic Books]], New York, 1964, page 12
*{{cite journal|author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[Ronald L. Graham|Graham, R. L.]]|title=Irregularities in the distributions of finite sequences
| journal = [[Journal of Number Theory]]|volume=2|year=1970|pages=152–161|id={{MathSciNet|id=0269605}}|doi=10.1016/0022-314X(70)90015-6}}
* M. Warmus, "A Supplementary Note on the Irregularities of Distributions", ''[[Journal of Number Theory]]'' 8, 260–263, 1976.

{{DEFAULTSORT:Irregularity Of Distributions}}
[[Category:Fractions]]

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174.30.131.200: /* Equation in the extended complex plane */