$\left[\begin{array}{l}[\text { Combinatorics (other) })] \\ [\text { Extreme Principle (other) })]\end{array}\right.$

In how many ways can the numbers from 1 to 100 be permuted so that adjacent numbers differ by no more than 1?
Where can the number 1 be placed?

## Solution

Next to the number 1, only the number 2 can stand, so 1 must be at the edge. Suppose 1 is at the beginning. Then the next number is 2, the next is 3 (no other numbers can be next to 2), the next is 4, and so on. We get the arrangement $1, 2, \ldots, 99, 100$.

If 1 is at the end, then similarly, the arrangement is uniquely restored as 100, 99, ..., 2, 1.

## Answer

In two ways.