Solution
The puzzle's solution is no. 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
Is 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 of I obeys the congruence
where counts how often the second rule is applied.
Read more about this topic: MU Puzzle
Famous quotes containing the word solution:
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“The truth of the thoughts that are here set forth seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, the final solution of the problems. And if I am not mistaken in this belief, then the second thing in which the value of this work consists is that it shows how little is achieved when these problems are solved.”
—Ludwig Wittgenstein (1889–1951)
“All the followers of science are fully persuaded that the processes of investigation, if only pushed far enough, will give one certain solution to each question to which they can be applied.... This great law is embodied in the conception of truth and reality. The opinion which is fated to be ultimately agreed to by all who investigate is what we mean by the truth, and the object represented in this opinion is the real.”
—Charles Sanders Peirce (1839–1914)