Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.
1. **Understanding the Problem:**
   We need to find the greatest natural number \( n \) such that for any partition of \( n \) into positive integers, there exists a fleet of ships on a \( 10 \times 10 \) grid where the summands are exactly the numbers of squares contained in individual ships. The ships must be vertex-disjoint.

2. **Initial Observation:**
   The grid is \( 10 \times 10 \), which contains \( 100 \) unit squares. We need to ensure that the ships are vertex-disjoint, meaning no two ships can share a common vertex.

3. **Partitioning the Grid:**
   Consider partitioning the \( 10 \times 10 \) grid into \( 2 \times 2 \) squares. Each \( 2 \times 2 \) square contains 4 unit squares. There are \( \frac{100}{4} = 25 \) such \( 2 \times 2 \) squares in the grid.

4. **Maximum Number of Ships:**
   If we place one ship in each \( 2 \times 2 \) square, we can have 25 ships, each occupying 4 unit squares. This configuration ensures that no two ships share a common vertex.

5. **Verification for \( n = 25 \):**
   For \( n = 25 \), we can represent \( 25 \) as a sum of 25 ones: \( 1 + 1 + \cdots + 1 = 25 \). This means we need 25 ships, each containing 1 unit square. However, this is not possible because placing 25 ships, each containing 1 unit square, would require 25 unit squares, and ensuring they are vertex-disjoint is not feasible in a \( 10 \times 10 \) grid.

6. **Verification for \( n = 24 \):**
   For \( n = 24 \), we can represent \( 24 \) as a sum of 24 ones: \( 1 + 1 + \cdots + 1 = 24 \). This means we need 24 ships, each containing 1 unit square. This is also not feasible for the same reason as above.

7. **Verification for \( n = 25 \) with Different Partitions:**
   For \( n = 25 \), consider different partitions such as \( 4 + 4 + \cdots + 4 + 1 = 25 \). We can place six \( 2 \times 2 \) ships and one ship containing 1 unit square. This configuration is feasible and ensures that the ships are vertex-disjoint.

8. **Generalization:**
   For any \( 2k \times 2m \) grid, the maximum number of vertex-disjoint ships is \( km \). For a \( 10 \times 10 \) grid, \( k = 5 \) and \( m = 5 \), so the maximum number is \( 5 \times 5 = 25 \).

Conclusion:
The greatest natural number \( n \) such that for any partition of \( n \) into positive integers, there exists a fleet of ships on a \( 10 \times 10 \) grid where the summands are exactly the numbers of squares contained in individual ships, is \( 25 \).

The final answer is \( \boxed{ 25 } \).