Take away my first letter and I remain unchanged; take away my second letter and I remain unchanged; take away my third letter and I remain unchanged; take away all my letters and still I remain exactly the same.
Source: Henry Dudeney, of course.
The answer to the riddle is "a postman"
The "letters" in the riddle denote the letters that he is carrying for delivery.
Of course, even after changing (or delivering) some or all the letters, the individual will still remain a postman.
Riddle Resolution
