Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq  m \leq  n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
1. **Constructing a Complete Sequence:**
   - We need to construct a sequence of \( n \) real numbers such that for every integer \( m \) with \( 1 \leq m \leq n \), either the sum of the first \( m \) terms or the sum of the last \( m \) terms is an integer.
   - Let's start by defining the sequence for \( n = 8 \) as an example:
     - Fill the first position with any integer, say \( a \).
     - Fill the two rightmost unoccupied cells with \( \frac{1}{2} \).
     - Fill the two leftmost unoccupied cells with \( \frac{1}{2} \).
     - Alternate between the leftmost and rightmost extremes until only one cell remains.
     - Fill that remaining cell with any integer.

   - For \( n = 8 \), the sequence would be constructed as follows:
     \[
     a, \_, \_, \_, \_, \_, \_, \_
     \]
     \[
     a, \_, \_, \_, \_, \_, \frac{1}{2}, \frac{1}{2}
     \]
     \[
     a, \frac{1}{2}, \frac{1}{2}, \_, \_, \_, \frac{1}{2}, \frac{1}{2}
     \]
     \[
     a, \frac{1}{2}, \frac{1}{2}, \_, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}
     \]
     \[
     a, \frac{1}{2}, \frac{1}{2}, a, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}
     \]

   - It is easy to see that this works because for any \( m \), either the sum of the first \( m \) terms or the sum of the last \( m \) terms will be an integer.

2. **Proving that 2 is the Minimum:**
   - Let \( x_i \) represent a real number that is not an integer, and \( a_i \) represent an integer.
   - The first or last cell must be an integer (let's assume the first cell is an integer, \( a_0 \)).
   - This forces us to fill the last two cells with \( x_1 \) and \( a_1 - x_1 \), because the sum of the first two cells cannot be an integer unless the second cell itself is an integer.
   - This process continues, forcing the 2nd and 3rd cells to be filled with \( x_2 \) and \( a_2 - x_2 \), and so on.
   - Eventually, we are left with one cell. The remaining \( n-1 \) cells sum to an integer because:
     \[
     a_0 + \sum_{j=1}^{i} (a_j - x_j) + \sum_{j=1}^{i} x_j = \sum_{j=0}^{i} a_j
     \]
     which is an integer.
   - The last cell must be filled with an integer because the sum of all \( n \) cells must be an integer.

   - Therefore, the minimum number of integers in a complete sequence of \( n \) numbers is 2.

The final answer is \(\boxed{2}\).