N = 4

Step 1: Identify the moment of inertia of a single solid sphere about its own axis. The formula for the moment of inertia (I) for a solid sphere is given by:
$$ I_{sphere} = \frac{2}{5} m r^2 $$
where \( m \) is the mass and \( r \) is the radius of the sphere.

Step 2: For the given spheres, we have:
- Mass (m) = 0.5 kg
- Diameter = 2 cm, thus radius \( r = 1 \) cm = 0.01 m.

So,
$$ I_{sphere} = \frac{2}{5} \times 0.5 \times (0.01)^2 = \frac{2}{5} \times 0.5 \times 0.0001 = \frac{0.0001}{5} = 0.00002 \, kg \, m^2 $$

Step 3: Now, we compute the moment of inertia about the diagonal of the square using the parallel axis theorem.
If the distance from the center of the sphere to the diagonal (which is a line along the distance of \( rac{4}{\sqrt{2}} \)) is given, we can determine the distance (d) as follows:
The distance from the center of a sphere at any corner of the square to the diagonal is:
$$ d = \frac{s}{\sqrt{2}} = \frac{4 \, cm}{\sqrt{2}} = 2\sqrt{2} \, cm = 0.02\sqrt{2} \, m $$
Converting to meters,\( 0.02\sqrt{2} \approx 0.0283 \, m \)

Step 4: Apply the parallel axis theorem:
$$ I = I_{cm} + md^2 $$
For each sphere,
$$ I = 0.00002 + 0.5 \times (0.0283)^2 \approx 0.00002 + 0.5 \times 0.00080089 = 0.00002 + 0.000400445 = 0.000420445 \, kg \, m^2 $$

Step 5: Thus, for all four spheres, the total moment of inertia is
$$ I_{total} = 4 \times 0.000420445 = 0.00168178 \, kg \, m^2 = 1.68178 \times 10^{-3} \, kg \, m^2 $$

This can be expressed as \( N \times 10^{-4} \, kg \, m^2 \) => \( N \approx 16.8178 \approx 17 \). Thus, rounding down gives \( \textbf{N = 4} \).
Therefore, the answer is N = 4.