The equation for a wave traveling on a string is given by $ y = \sin(20\pi x + 10\pi t) $, where $ x $ is the distance and $ t $ is the time in SI units. To find the minimum distance between two points having the same oscillating speed, we use the concept that this distance is half the wavelength $\left(\frac{\lambda}{2}\right)$.

To find the wavelength $\lambda$:

$ \lambda = \frac{2\pi}{k} $

Here, the wave number $ k = 20\pi $. Thus,

$ \lambda = \frac{2\pi}{20\pi} = \frac{1}{10} \text{ m} = 10 \text{ cm} $

Thus, the minimum distance between two points having the same speed is:

$ \text{Distance} = \frac{\lambda}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} $