Guess the color of the cap - extended

Question

I had here (Guess the color of the cap) the following problem:

I have 3 persons which either wear a white or a black cap. They can only see the color of the other caps, but not their own. White and black caps are eqally likely. As a team, they play a game of guessing their own cap color. If they will win, all of them have to guess correctly their own cap color. Once the game begins, they cannot communicate the color of the other two caps.

Now, what if there aren't 3 persons, but $n$ persons - and not 2 colors, but $r$ distinct colors (each color equally likely)? With which strategy do, with 100% certainly, at least $n-1$ answer correctly?

Thanks for any hint!

Answer 1

Hint:

Step 1: Person $1$ makes a guess. This might be right or wrong.

Step 2 thru n: All other people will guess correctly, using information from Person $1$'s guess.

Bonus Hint: $\mathbb{Z}/r\mathbb{Z}$, i.e. arithmetic modulo $r$.