The M-I-U Puzzle

Just finished reading Chapter 1 of Gödel, Escher & Bach: An Eternal Golden Braid by D. R. Hofstadter, in which the author presents the following puzzle. The puzzle has been around for a while, so I ask those of you who know the answer to not reveal it in the comments sections. I will reveal the answer when I get to the appropriate section in the book.

The M-I-U Puzzle

It’s a puzzle utilizing only three letters in the alphabet. Starting with the axiom “MI”, can you form the theorem “MU”? You must follow these rules:

1) If you possess a string with “I” as the last letter, you can add a “U”.
Example: From “MI” you can get “MIU”
From “MUIIUII” you can get “MUIIUIIU”
From “MIIU” you can’t get diddly squat using rule 1

2) Suppose you have “Mx” (where “x” represents any string of I’s and U’s). You may add another “x” at the end to create “Mxx”.
Example: From “MIU” you may get “MIUIU”
From “MUM” you may get “MUMUM”
From “MU” you may get “MUU”

3) If the string “III” occurs, you may replace it with a “U”
Example: From “MIIIU” you could make “MUU”
From “MIIII” you could make either “MIU” or “MUI”
From MII you can’t get diddly squat using rule 3.
NOTE!!! You can’t go backwards and turn a “U” into and “III”

4) If the string “UU” occurs, you may drop it.
Example: From “MUUU” you could make “MU”
From “MUUI” you could make either “MI”

Go ahead folks, see if you can turn “MI” into “MU”. Have fun. Don’t cheat.