Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $


[i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]
To determine the largest natural number \( N \) such that every \( 5 \times 5 \) array consisting of pairwise distinct natural numbers from \( 1 \) to \( 25 \) contains a \( 2 \times 2 \) subarray of numbers whose sum is at least \( N \), we need to analyze the properties of such arrays and subarrays.

1. **Sum of All Elements**:
   The sum of all elements in a \( 5 \times 5 \) array containing numbers from \( 1 \) to \( 25 \) is:
   \[
   \sum_{i=1}^{25} i = \frac{25 \cdot 26}{2} = 325
   \]

2. **Average Sum of a \( 2 \times 2 \) Subarray**:
   There are \( 16 \) possible \( 2 \times 2 \) subarrays in a \( 5 \times 5 \) array. If we denote the sum of the elements in the \( i \)-th \( 2 \times 2 \) subarray by \( S_i \), then:
   \[
   \sum_{i=1}^{16} S_i = 4 \cdot 325 = 1300
   \]
   This is because each element in the \( 5 \times 5 \) array is counted exactly four times when summing over all \( 2 \times 2 \) subarrays.

3. **Average Value of \( S_i \)**:
   The average value of \( S_i \) is:
   \[
   \frac{1300}{16} = 81.25
   \]
   Since \( S_i \) must be an integer, the average sum of a \( 2 \times 2 \) subarray is approximately \( 81 \).

4. **Minimum Sum of a \( 2 \times 2 \) Subarray**:
   To find the minimum possible sum of a \( 2 \times 2 \) subarray, consider the smallest possible values for the elements in such a subarray. The smallest \( 2 \times 2 \) subarray would consist of the numbers \( 1, 2, 6, \) and \( 7 \):
   \[
   1 + 2 + 6 + 7 = 16
   \]
   However, this is not relevant to our problem since we are looking for the largest \( N \) such that every \( 2 \times 2 \) subarray has a sum of at least \( N \).

5. **Constructing a \( 5 \times 5 \) Array**:
   We need to construct a \( 5 \times 5 \) array where the sum of every \( 2 \times 2 \) subarray is at least \( 45 \). Consider the following array:
   \[
   \begin{bmatrix}
   17 & 10 & 20 & 7 & 23 \\
   16 & 1 & 14 & 3 & 12 \\
   18 & 9 & 21 & 6 & 24 \\
   15 & 2 & 13 & 4 & 11 \\
   19 & 8 & 22 & 5 & 25 \\
   \end{bmatrix}
   \]
   Let's check the sums of the \( 2 \times 2 \) subarrays:
   \[
   \begin{array}{cccc}
   44 & 45 & 44 & 45 \\
   44 & 45 & 44 & 45 \\
   44 & 45 & 44 & 45 \\
   44 & 45 & 44 & 45 \\
   \end{array}
   \]
   Each \( 2 \times 2 \) subarray has a sum of at least \( 44 \) and at most \( 45 \).

6. **Conclusion**:
   Since we have constructed a \( 5 \times 5 \) array where the sum of every \( 2 \times 2 \) subarray is at least \( 45 \), we conclude that the largest \( N \) is \( 45 \).

The final answer is \( \boxed{45} \).