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Permuting the cube	---	-v-	transition none

Permuting the cube

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Hi - I like cubes 👋

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Metamagical Themas - D. Hofstadter

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Permutations

• Possible sequences / arrangements
• $$\begin{pmatrix}1 & 2 & 3\\1 & 3 & 2\end{pmatrix}$$
• $$\begin{pmatrix}1 & 3 & 2\end{pmatrix}$$

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The symmetric group

• $S_n$ - in this case $S_{54}$
• identity
• inverse
• they combine

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The R permutation

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The R permutation

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Orienting the cube

• Let's have some permutations to orient the cube
• They're not a direct help, but can contribute

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The X permutation

$\rightarrow$ We get $X^{-1}$, also written $X'$ from this ;)

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The Y permutation

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The Z permutation

• Let's just do this: $X\cdot Y\cdot X'$

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The F permutation

• Let's just do this: $Z\cdot R\cdot Z'$

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The whole bunch

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Towards solving

• each shuffle of a cube corresponds to a permutation
• we search for a sequence of permutations that combines to the solution

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example shuffle

Let's take an example:

$$ S = L\cdot D\cdot L'\cdot D'\cdot L\cdot D\cdot L' $$

and it's inverse:

$$ S' = L\cdot D'\cdot L'\cdot D\cdot L\cdot D'\cdot L' $$

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what $S$ looks like:
$S = L\cdot D\cdot L'\cdot D'\cdot L\cdot D\cdot L'$

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Let's add a $U$:
$S\cdot U$

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And back again:
$S\cdot U \cdot S'\cdot U'$

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we got us a commutator

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Thanks for having me

• Slides: github.com/runjak/2020-06-06.enthusiasticon
• Twitter: @sicarius
• Happy to chat on zulip 🐋✨