Suppose you need to authenticate yourself to a program with the password - but the program's source code is public, the program doesn't have access to any private information and all your communications with the program are public. The only privacy you have is the information that you have but don't send.
No fixed password could work because the first time you used it, everyone would know it, and all subsequent runs of the program would be compromised. One partial solution would be to have a different password every second. But naive password generation wouldn't work, since both the source code of the program and any information it has access to are known.
Is there a (secure) way for the program to quickly generate password hashes, so that a trusted user with the right information could quickly generate passwords matching those hashes? This would be difficult to attack iff it is hard to predict the next password given the next hash and all the previous passwords. It is easy to make it so that nobody can come up with passwords (just randomly generate hashes), but making it actually possible for one person to generate passwords (quickly) given a little secret information seems much harder to me.