Determine all natural numbers $n$ so that numbers $1, 2,... , n$ can be placed on the circumference of a circle and  for each natural number $s$ with $1\le s \le \frac12n(n+1)$ , there is a circular arc which has the sum of all numbers in that arc to be $s$.
To determine all natural numbers \( n \) such that the numbers \( 1, 2, \ldots, n \) can be placed on the circumference of a circle and for each natural number \( s \) with \( 1 \le s \le \frac{1}{2}n(n+1) \), there is a circular arc which has the sum of all numbers in that arc equal to \( s \), we consider two cases: \( n \) is even and \( n \) is odd.

1. **Case 1: \( n \) is even**

   - Let \( n = 2k \) for some integer \( k \).
   - Arrange the numbers \( 1, 2, \ldots, n \) on the circle such that each pair of numbers that add up to \( n+1 \) are placed next to each other. Specifically, place the pairs \( (1, n), (2, n-1), \ldots, (k, k+1) \) consecutively around the circle.
   - To find a circular arc whose sum of numbers in that arc is \( a(n+1) + b \) where \( 0 \le b \le n \):
     - Start the arc from the number \( b \).
     - Move in the direction opposite to \( b \)'s pair.
     - Stop after passing exactly \( a \) pairs.
     - If \( b = 0 \), simply find the arc covering exactly \( a \) pairs.

2. **Case 2: \( n \) is odd**

   - Let \( n = 2k + 1 \) for some integer \( k \).
   - Arrange the numbers \( 1, 2, \ldots, n \) on the circle such that each pair of numbers that add up to \( n \) are placed next to each other, and place \( n \) by itself without splitting any other pair. Specifically, place the pairs \( (1, n-1), (2, n-2), \ldots, (k, k+1) \) consecutively around the circle, with \( n \) placed separately.
   - To find a circular arc whose sum of numbers in that arc is \( a(n) + b \) where \( 0 \le b \le n \):
     - Start the arc from the number \( b \).
     - Move in the direction opposite to \( b \)'s pair.
     - Stop after passing exactly \( a \) pairs.
     - If \( b = 0 \), simply find the arc covering exactly \( a \) pairs.

By following these constructions, we can ensure that for any natural number \( s \) within the specified range, there exists a circular arc whose sum of numbers equals \( s \).

The final answer is all natural numbers \( \boxed{ n } \).