In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$.
[img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]
1. **Identify the properties of the shapes involved:**
   - The hexagon is regular, meaning all its sides and angles are equal.
   - Each square has a side length of 4 cm.
   - The centers of the squares, \(M\), \(N\), and \(P\), form the vertices of triangle \(MNP\).

2. **Determine the side length of the equilateral triangle \(MNP\):**
   - Since the squares are regular and their sides are parallel to the sides of the hexagon, the centers \(M\), \(N\), and \(P\) form an equilateral triangle.
   - The distance between the centers of two adjacent squares can be calculated by considering the geometry of the squares and the hexagon.

3. **Calculate the distance between the centers of two adjacent squares:**
   - Each square has a side length of 4 cm.
   - The distance between the centers of two adjacent squares is the sum of the side of the square and the height of the equilateral triangle formed by the centers of the squares.
   - The height of an equilateral triangle with side length \(s\) is given by \(\frac{\sqrt{3}}{2} s\).

4. **Calculate the height of the mini equilateral triangles inside the squares:**
   - The height of one of these mini triangles is 2 cm (half the side length of the square).
   - The side length of these mini triangles is \(\frac{4}{\sqrt{3}}\).

5. **Calculate the side length of \(\triangle{MNP}\):**
   - The side length of \(\triangle{MNP}\) is the sum of the side length of the square and twice the height of the mini equilateral triangles.
   - Therefore, the side length of \(\triangle{MNP}\) is \(4 + 2 \times \frac{4}{\sqrt{3}} = 4 + \frac{8}{\sqrt{3}} = 4 + \frac{8\sqrt{3}}{3}\).

6. **Calculate the perimeter of \(\triangle{MNP}\):**
   - The perimeter of \(\triangle{MNP}\) is three times the side length.
   - Thus, the perimeter is \(3 \left(4 + \frac{8\sqrt{3}}{3}\right) = 12 + 8\sqrt{3}\).

7. **Express the perimeter in the form \(a + b\sqrt{3}\):**
   - Here, \(a = 12\) and \(b = 8\).

8. **Compute the value of \(a + b\):**
   - \(a + b = 12 + 8 = 20\).

The final answer is \(\boxed{20}\).