The Stargate Switch

Dana C. Ernst

http://danaernst.com

FAMUS

February 1, 2013

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Stargate

Stargate SG-1, "Holiday", Season 2

The Stargate Switch Problem

Here are the swaps that take place during the episode:

The question is:

Can the machine be used to put everyone's mind back in the proper body if we are not allowed to use the machine on the same pair of bodies more than once?

The Generalized Stargate Switch

Note

This is not as general as the problem posed in Futurama. In that case, we needed to introduce outsiders to solve the problem. But here we can solve the problem without outsiders (as long as we have at least 2 pairs of bodies).

Let's use some math!

The symmetric group

We can model this problem in terms of permutations (i.e., rearrangements) of numbers. It turns out that the set of all permutations of the numbers 1 through $n$ forms a "group" under composition.

A group is a set with an associative binary operation satisfying:

The group of permutations of $\{1,2,\ldots, n\}$ is called the symmetric group and is denoted by $S_n$.

Permutation diagrams

One way of representing an element of the symmetric group is via permutation diagrams, which we illustrate by way of example.

Example

Diagram1

Permutation diagrams (continued)

Let's try multiplying.

Example

Let $\alpha$ and $\beta$ be as on previous slide. Then

Diagram2

But on the other hand

Diagram3

We see that products of permutations do not necessarily commute (order matters).

Some Comments

Cycle Notation

We need to introduce a more efficient way of encoding our permutations. One such method is called cycle notation.

Example

Diagram6

Cycle Notation (continued)

Let's try multiplying using cycle notation. Note: We will multiply right to left (like function composition).

Example

Consider $\alpha$ and $\beta$ in $S_{5}$ as in the previous examples. Then

\[\alpha \beta=(1, 2, 3, 4, 5)(2, 4, 3)=(1, 2, 5).\]

More Comments

Solution to Original Stargate Switch

Evans and Huang from the University of California, San Diego recently proved the following theorem. [arXiv]

Theorem

The following permutation solves the original Stargate Switch Problem.

\[\mu=(2, 4)(1, 3)(2, 3)(1, 4)\]

Moreover, this solution is optimal in the sense that no solution with fewer than 4 moves will work.

Proof for Original Stargate Switch

Proof

First, easy to check that $\mu\sigma$ yields the identity:

\[\mu\sigma=(2, 4)(1, 3)(2, 3)(1, 4)\circ (1,2)(3,4)=\text{identity permutation}\]

Why is $\mu$ optimal? Well, it's easy to check that 2 transpositions (not reusing the originals and only using $1,2,3,4$) won't work. The reason why $\mu$ cannot only have 3 cycles is because of a general fact that says that the identity permutation cannot be equal to a product of an odd number of transpositions (sometimes called the Parity Theorem).

Solution to Generalized Stargate Switch

Evans and Huang also proved the following more general theorem. [arXiv]

Theorem

Let $m>1$. Then the Generalized Stargate Switch permutation \[\sigma=(1, 2)(3, 4)\cdots (2m-1, 2m)\] can be undone by a product $\mu$ of $N(m)$ distinct transpositions, where for $j=4i$ \[\mu=\begin{cases}\tau \prod_{i=2}^{(m-1)/2} (j, j+2)(j-1, j+1)(j, j+1)(j-1, j+2), & \text{if }m\text{ is odd}\\ \prod_{i=1}^{m/2}(j-1, j)(j-3, j-1)(j-2, j-1)(j-3, j), & \text{if }m\text{ is even}\end{cases}\] with \[\tau=(1,3)(1,6)(4,5)(4,6)(3,5)(2,5)(1,5).\] and \[N(m)=\begin{cases}2m+1, & \text{if }m\text{ is odd (and greater than 1),}\\ 2m, & \text{if }m\text{ is even.}\end{cases}\] (The empty product for $m=3$ is interpreted as the identity permutation.) Moreover, $N(m)$ is optimal.

Examples of Solutions

The previous formula looks pretty scary, but it's not too bad.

Example

When $m=3$, we have

\[\mu= \tau=(1,3)(1,6)(4,5)(4,6)(3,5)(2,5)(1,5),\]

which consists of $N(3)=7$ transpositions.

When $m=4$, we have

\[\mu=(2,4)(1,3)(2,3)(1,4)(6,8)(5,7)(6,7)(5,8),\]

which consists of $N(4)=8$ transpositions.

Resources and More Information

I utilized the Wikipedia entries for both Stargate and Stargate SG-1.

If you'd like to know more, check out the following papers:

You might also be interested in one of my recent talks about the Futurama Theorem.

These slides were made using deck.js and MathJax. In addition, I modified Christian Perfect's deck.js template and made use of Jason B. Hill's pdiag package to create the permutation diagrams.

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