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
,
…
,
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
,
…
,
x
N
such that for every n ∈ {1, ..., N} and every k ∈ {1, ..., n} there is some i ∈ {1, ..., n} such that
k
−
1
n
≤
x
i
<
k
n
.
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:
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
In this example, considering for instance the first 5 numbers, we have
0
<
x
1
<
1
5
<
x
5
<
2
5
<
x
3
<
3
5
<
x
4
<
4
5
<
x
2
<
1.
