BdMO National 2016 Higher Secondary 

[u][b]Problem 4:[/b][/u]
Consider the set of integers $  \left \{ 1, 2, ......... , 100 \right \} $. Let $   \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum,

$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2  \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $.
To find the smallest possible value of the sum \( S = \left| x_2 - x_1 \right| + \left| x_3 - x_2 \right| + \cdots + \left| x_{100} - x_{99} \right| + \left| x_1 - x_{100} \right| \), we need to consider the arrangement of the integers \( \{1, 2, \ldots, 100\} \).

1. **Understanding the Problem:**
   The sum \( S \) represents the total distance traveled if we start at \( x_1 \), move to \( x_2 \), then to \( x_3 \), and so on, until we return to \( x_1 \). The goal is to minimize this total distance.

2. **Distance Between Consecutive Points:**
   The distance between two points \( x_i \) and \( x_{i+1} \) is given by \( \left| x_{i+1} - x_i \right| \). To minimize the sum of these distances, we should arrange the points such that the distances are as small as possible.

3. **Optimal Arrangement:**
   The smallest possible distance between any two points on the number line is 1. Therefore, to minimize the sum \( S \), we should arrange the integers in consecutive order. 

4. **Calculating the Sum for Consecutive Order:**
   If we arrange the integers in consecutive order, i.e., \( x_i = i \) for \( i = 1, 2, \ldots, 100 \), the sum \( S \) becomes:
   \[
   S = \left| 2 - 1 \right| + \left| 3 - 2 \right| + \cdots + \left| 100 - 99 \right| + \left| 1 - 100 \right|
   \]

5. **Evaluating the Sum:**
   Each term \( \left| x_{i+1} - x_i \right| \) for \( i = 1, 2, \ldots, 99 \) is equal to 1. Therefore, the sum of the first 99 terms is:
   \[
   \sum_{i=1}^{99} \left| x_{i+1} - x_i \right| = 99 \times 1 = 99
   \]
   The last term \( \left| x_1 - x_{100} \right| \) is:
   \[
   \left| 1 - 100 \right| = 99
   \]

6. **Total Sum:**
   Adding these together, we get:
   \[
   S = 99 + 99 = 198
   \]

Therefore, the smallest possible value of the sum \( S \) is \( 198 \).

The final answer is \( \boxed{198} \).