MathDB

A finite sequence is said to be disorderly, if no two terms of the sequence have their average in between them. For example,

(0, 2, 1)

is disorderly, for

1 = \frac{0+2}{2}

is not in between

0

and

2

, and the other averages

\frac{0+1}{2} = \frac{1}{2}

and

\frac{2+1}{2} = 1\frac{1}{2}

do not even occur in the sequence. Prove that for every

n \in \Bbb{N}

there is a disorderly sequence enumerating the numbers

0, 1,\ldots , n

without repetitions.

Problem Statement