Earlier quoted context omitted.
If you only remember 25% of the characters, then you have to ask for 75% of the characters to generate the full hash. Asking for 75% of the characters defeats the purported reasons for partial password auth.
Why? Now all the universal keyloggers don't work (because they remember which keys were pressed, but not which positions they refer to).
Besides which, the partial login isn't random either. It's always sequential. You'll never be asked to enter character 3, then 1, then 8. It will always be asked in the same order as the password string itself.