A sequence with \(N\) elements will be called a permutation of the \(N\) smallest natural numbers if it contains all natural numbers from \(1\) to \(N\).

It is known that the sequence \(\left\{ a_{i} \right\}\) is a permutation of the \(n\) smallest natural numbers (\(n>3\)).

The elements of the sequence \(\left\{ b_{i} \right\}\) (\(1 \leq i \leq n-1\)) are computed by the formula \(b_{i}=\left|a_{i+1}-a_{i}\right|\).

The elements of the sequence \(\left\{ c_{i} \right\}\) (\(1 \leq i \leq n-2\)) are computed by the formula \(c_{i}=\left|b_{i+1}-b_{i}\right|\).

Prove that \(\left\{ b_{i} \right\}\) and \(\left\{ c_{i} \right\}\) cannot both simultaneously be permutations of the \(n-1\) and \(n-2\) smallest natural numbers, respectively!

Skaitļu virkni, kurā ir \(N\) elementi, sauksim par \(N\) mazāko naturālo skaitļu permutāciju, ja tajā atrodami visi naturālie skaitļi no \(1\) līdz \(N\).

Zināms, ka virkne \(\left\{ a_{i} \right\}\) ir \(n\) (\(n>3\)) mazāko naturālo skaitļu permutācija.

Virknes \(\left\{ b_{i} \right\}\) (\(1 \leq i \leq n-1\)) elementus aprēķina pēc formulas \(b_{i}=\left|a_{i+1}-a_{i}\right|\).

Virknes \(\left\{ c_{i} \right\}\) (\(1 \leq i \leq n-2\)) elementus aprēķina pēc formulas \(c_{i}=\left|b_{i+1}-b_{i}\right|\).

Pierādīt, ka \(\left\{ b_{i} \right\}\) un \(\left\{ c_{i} \right\}\) vienlaikus abas nevar būt attiecīgi \(n-1\) un \(n-2\) mazāko naturālo skaitļu permutācijas!