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:

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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