# Strings in the MIU System

Logic Level 1

In the $$MIU$$ system, you always start with the string $$MI$$ and then form new strings by applying any of following rules any number of times:

 Rules Examples 1. If your string ends with $$I$$, you can add a $$U$$ at the end. $$MI\longrightarrow MIU$$ 2. You can double the entire string following $$M.$$ $$MIU\longrightarrow MIUIU$$ 3. Three consecutive $$I$$'s can be replaced with a single $$U.$$ $$MIIIU\longrightarrow MUU$$ 4. Two consecutive $$U$$'s can be removed from the string. $$MUU\longrightarrow M$$

Which of the following strings can be derived from $$MI$$ using these rules?

Note: None of the rules can be used in the opposite way; for example, you aren't allowed to derive $$MUUI$$ from $$MI$$ by using a reversed version of rule 4.

Source: The $$MIU$$ puzzle was introduced by Douglas Hofstadter in his magisterial work Gödel, Escher, Bach.

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