1. let $n \geq 2$ be a natural number. Depending on $n$, find the largest natural number $d$ such that there exists a permutation $a_{1}, a_{2}, \ldots, a_{n}$ of the numbers $1,2, \ldots, n$ with

$$
\left|a_{i}-a_{i+1}\right| \geq d, \quad \text { for } i=1,2, \ldots, n-1
$$

## Solution
We first consider the case where $n$ is odd, i.e. $n=2 k+1$.

Then for every $m \in\{1,2, \ldots, 2 k+1\}$ surely $|k+1-m| \leq k$ applies, i.e. $d \leq k$. With the following permutation, we see that $d=k$ is also possible:

$$
\begin{aligned}
a_{2 i+1} & =k+1+i, \text { for } i=0,1, \ldots, k \\
a_{2 i} & =i, \text { for } i=1,2, \ldots, k
\end{aligned}
$$

Now let $n$ be even, i.e. $n=2 k$.

Now for all $m \in\{1,2, \ldots, 2 k\}$ the estimate $|k-m| \leq k$ applies, i.e. $d \leq k$. With the following permutation, we see that $d=k$ is also possible:

$$
\begin{aligned}
a_{2 i+1} & =k+1+i, \text { for } i=0,1, \ldots, k-1 \\
a_{2 i} & =i, \text { for } i=1,2 \ldots, k
\end{aligned}
$$