How to make a color black and white?

Question

Design

By what rule can any color be converted to black and white? This can be seen in Photoshop when using the Grayscale mode.
PS There were thoughts about the average value, but it did not work out.

Answer 1

1) RGB -> HSB
2) set saturation to zero
3) convert HSB -> RGB back

Answer 2

Different colors have different susceptibility of people. eye, respectively, can not be averaged.
If you mean the formula, then here is one of the options:
G = 0.21 * R + 0.71 * G + 0.07 * B