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In yesterday's series, the futurama posed a rather interesting puzzle — I couldn't help but hold out and disassemble it here.

So, the plot is as follows: The professor invented a machine for the exchange of bodies, which, as it turned out, works only in one direction. After several permutations, the heroes found themselves in a difficult situation in which they had to figure out a way to return to their bodies. This is where pure mathematics will help us.

So let's get started. Let be Is a cycle of length*k* on the set *[n] = {1 ... n}* . Without loss of generality, we write:

Now suppose*(a, b)* is a transposition that interchanges the contents of *a* and *b* .

By assumption obtained by using certain substitutions over*[n]* .

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We introduce two "new bodies"*{x, y}* and write

For any*i = 1, ... k* we write as a series of permutations:

Note that transpositions change an element from*[n]* with some element from *{x, y}* , therefore all transpositions differ from the ones that formed the original permutation , and also from transposition *(x, y)* . By a simple check we get:

In this way, inverts a cycle of length*k* , leaving *x* and *y* interchanged without using the transposition *(x, y)* .

Now let - random substitution; it splits into a composition of independent cycles, each of which can be inverted using the algorithm obtained above, after which, if necessary, you can swap*x* and *y* using the transposition *(x, y)* .

So that. And you say that the discrete has no IRL applications.

So, the plot is as follows: The professor invented a machine for the exchange of bodies, which, as it turned out, works only in one direction. After several permutations, the heroes found themselves in a difficult situation in which they had to figure out a way to return to their bodies. This is where pure mathematics will help us.

So let's get started. Let be Is a cycle of length

Now suppose

By assumption obtained by using certain substitutions over

')

We introduce two "new bodies"

For any

Note that transpositions change an element from

In this way, inverts a cycle of length

Now let - random substitution; it splits into a composition of independent cycles, each of which can be inverted using the algorithm obtained above, after which, if necessary, you can swap

So that. And you say that the discrete has no IRL applications.

Source: https://habr.com/ru/post/102263/