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This question is about a puzzle from the book of Raymond Smullyan: The Lady or the Tiger? The description of the puzzle starts in Chapter 8, p. 103, and here is the important part (copy-pasted from Bruce Ediger's webpage):

Property Q: For any combination x, the combination QxQ is specially related to x.

Property L: If x is specially related to y, then Lx is specially related to Qy.

Property V (the reversal property: If x is specially related to y, then Vx is specially related to the reverse of y.

Property R (the repetition property: If x is specially related to y, then Rx is specially related to yy.

Property Sp: If x is specially related to y, then if x jams the lock, y is neutral, and if x is neutral, then y jams the lock.

Smullyan provides a key of length 10 on p. 163, however, this is not the shortest, I know a key of length 8. Is this the shortest? Spoiler alert: You can find this key in my comment to the accepted answer.)

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    $\begingroup$ For multiple reasons, including link rot, it is usually a good practice to actually reproduce the question you're asking. $\endgroup$ – Michaël Mar 20 '17 at 10:25
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    $\begingroup$ @M Agree, done! $\endgroup$ – domotorp Mar 20 '17 at 10:57
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Edited in recognition of closed-mindedness.

My brute force search shows no keys shorter than 10. Here are the only keys of length 10 and 11 respectively:

RVLVQRVLVQ VRLVQVRLVQ

VLRVQVLRVQQ VLVRQVLVRQQ

Curiously, there are no keys of length 12.

The only word of length 7 that does not crash under iteration is RRQRRQQ, and it evolves unboundedly. There are 74 words of length 8 that grow to over 30 letters, I think none of them cycles, and there are two eventually cycling words, one you gave and the other its pair RQVRLVQQ. The first time an odd period greater than 1 appears is at length 12, these are the originating words (all end up with period 3):

RQQVLLRLVQQQ

RLQVLLRLVQQQ

RQLVLLLRVQQQ

RQVLLLVLRQQQ

RQLVLLLVRQQQ

RQLVLLLRVLQQ

RQVLLLVLRLQQ

RQLVLLLVRLQQ

RQLLVLLRLVQQ

RQVLLLVQQRQQ

RQVLLLVLQRQQ

RQLVLLLVQRQQ

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    $\begingroup$ Ok, then I give here my key of length 8 (which can be extended to length 9 keys), spoiler alert! RQRVLVQQ $\endgroup$ – domotorp Feb 4 '10 at 16:56
  • $\begingroup$ You're right, I edited my answer accordingly. $\endgroup$ – Thorny Feb 5 '10 at 9:46
  • $\begingroup$ Wow, nice list, thank you! But why do you say that RRQRRQQ evolves unboundedly? It seems to me that it crashes in the 3rd step. $\endgroup$ – domotorp Feb 5 '10 at 12:03
  • $\begingroup$ Because I only checked iterations up to length 30, and the maximal achieved length of that word is 32. Sorry. $\endgroup$ – Thorny Feb 5 '10 at 12:19

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