Put the rectangle to have corners $(0, 0)$ and $(100, \sqrt{3})$ A covering exists by placing napkins with center $\left(n + 0.5, \frac{\sqrt{3}}{2}\right)$ for $n = 0, 1, \dots, 99$ which each cover a $\sqrt{3} \times 1$ rectangle so $100$ napkins work. On the other hand, if we consider the points \[ \left(0.5, \frac{\sqrt{3}}{2} + 0.1434\right), \left(1.5, \frac{\sqrt{3}}{2} - 0.1434\right), \left(2.5, \frac{\sqrt{3}}{2} + 0.1434\right) \]then no napkin covers two of these points so $100$ napkins are needed to cover all of them.