The **MU puzzle** is a puzzle stated by Douglas Hofstadter and found in *Gödel, Escher, Bach*. As stated, it is an example of a Post canonical system and can be reformulated as a string rewriting system.

## The puzzle

Suppose there are the symbols `M`, `I`, and `U` which can be combined to produce strings of symbols. The MU puzzle asks one to start with the "axiomatic" string `MI` and transform it into the string `MU` using in each step one of the following transformation rules:^{[1]}^{[2]}

**Nr.****Formal rule**^{[note 1]}**Informal explanation****Example**1. *x*`I`→ *x*`IU`Add a `U`to the end of any string ending in`I``MI`to `MIU`2. `M`*x*→ `M`*xx*Double the string after the `M``MIU`to `MIUIU`3. *x*`III`*y*→ *x*`U`*y*Replace any `III`with a`U``MUIIIU`to `MUUU`4. *x*`UU`*y*→ *xy*Remove any `UU``MUUU`to `MU`

## Solution

The puzzle cannot be solved: it is impossible to change the string `MI` into `MU` by repeatedly applying the given rules.

In order to prove assertions like this, it is often beneficial to look for an invariant; that is, some quantity or property that doesn't change while applying the rules.

In this case, one can look at the total number of `I` in a string. Only the second and third rules change this number. In particular, rule two will double it while rule three will reduce it by 3. Now, the *invariant property* is that the number of `I` is not divisible by 3:

- In the beginning, the number of
`I`s is 1 which is not divisible by 3. - Doubling a number that is not divisible by 3 does not make it divisible by 3.
- Subtracting 3 from a number that is not divisible by 3 does not make it divisible by 3 either.

Thus, the goal of `MU` with zero `I` cannot be achieved because 0 *is* divisible by 3.

In the language of modular arithmetic, the number *n* of `I` obeys the congruence

- $n\equiv {2}^{a}\not\equiv 0\phantom{\rule{1em}{0ex}}.\phantom{\rule{thinmathspace}{0ex}}$

where *a* counts how often the second rule is applied.

## Arithmetization

Chapter XIX of *Gödel, Escher, Bach* gives a mapping of the MIU system to arithmetic, as follows. First, every MIU-string can be translated to an integer by mapping the letters `M`, `I`, and `U` to the numbers 3, 1, and 0, respectively. (For example, the string `MIUIU` would be mapped to 31010.)

Second, the single axiom of the MIU system, namely the string `MI`, becomes the number 31.

Third, the four formal rules given above become the following:

**Nr.****Formal rule**^{[note 2]}**Example**1. *k*× 10 + 1→ 10 × ( *k*× 10 + 1)31 → 310 ( *k*= 3)2. 3 × 10 ^{m}+*n*→ 10 ^{m}× (3 × 10^{m}+*n*) +*n*310 → 31010 ( *m*= 2,*n*= 10)3. *k*× 10^{m + 3}+ 111 × 10^{m}+*n*→ *k*× 10^{m + 1}+*n*3111011 → 30011 ( *k*= 3,*m*= 3,*n*= 11)4. *k*× 10^{m + 2}+*n*→ *k*× 10^{m}+*n*30011 → 311 ( *k*= 3,*m*= 2,*n*= 11)

(**NB:** The rendering of the first rule above differs superficially from that in the book, where it is written as "[i]f we have made 10*m* + 1, then we can make 10 × (10*m* + 1)". Here, however, the variable *m* was reserved for use in exponents of 10 only, and therefore it was replaced by *k* in the first rule. Also, in this rendering, the arrangement of factors in this rule was made consistent with that of the other three rules.)

## Relationship to logic

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The MIU system illustrates several important concepts in logic by means of analogy.

It can be interpreted as an analogy for a formal system — an encapsulation of mathematical and logical concepts using symbols. The MI string is akin to a single axiom, and the four transformation rules are akin to rules of inference.

The MU string and the impossibility of its derivation is then analogous to a statement of mathematical logic which cannot be proven or disproven by the formal system.

It also demonstrates the contrast between interpretation on the "syntactic" level of symbols and on the "semantic" level of meanings. On the syntactic level, there is no knowledge of the MU puzzle's insolubility. The system does not *refer* to anything: it is simply a game involving meaningless strings. Working within the system, an algorithm could successively generate every valid string of symbols in an attempt to generate MU, and though it would never succeed, it would search forever, never deducing that the quest was futile. For a human player however, after a number of attempts, one soon begins to suspect that the puzzle may be impossible. One then "jumps out of the system" and starts to reason *about* the system, rather than working within it. Eventually, one realises that the system is in some way *about* divisibility by three. This is the "semantic" level of the system — a level of meaning that the system naturally attains. On this level, the MU puzzle can be seen to be impossible.

The inability of the MIU system to express or deduce facts about itself, such as the inability to derive MU, is a consequence of its simplicity. However, more complex formal systems, such as systems of mathematical logic, may possess this ability. This is the key idea behind Godel's Incompleteness Theorem.

## Pedagogical Uses

In her textbook, *Discrete Mathematics with Applications*, Susanna S. Epp uses the MU puzzle to introduce the concept of recursive definitions, and begins the relevant chapter with a quote from *GEB*.^{[3]}

## See also

## Notes

**^**Here,*x*and*y*are variables, standing for strings of symbols. A rule may be applied only to the whole string, not to an arbitrary substring.**^**Here,*k*and*m*stand for arbitrary natural numbers, and*n*is any natural number smaller than 10^{m}. Each rule of the form "*x*→*y*" should be read as "if we have made*x*we can make*y*." As the Example column illustrates, a rule may be applied only to an entire MIU-number, not to an arbitrary part of its decimal representation.

## References

**^**Justin Curry / Curran Kelleher (2007).*Gödel, Escher, Bach: A Mental Space Odyssey**. MIT OpenCourseWare.***^**Hofstadter, Douglas R. (1999) [1979],*Gödel, Escher, Bach: An Eternal Golden Braid*, Basic Books, ISBN 0-465-02656-7 Here: Chapter I.**^***Discrete Mathematics with Applications*, Third Edition, Brooks/Cole, 2004. Chapter 8.4, "General Recursive Definitions," p. 501.

## External links

- "Hofstadter's MIU System". Archived from the original on 4 March 2016. Retrieved 29 November 2016. An online interface for producing derivations in the MIU System.
- "MU Puzzle". Retrieved 13 May 2018. An online JavaScript implementation of the MIU production system.