School of Logic: a Collection of Puzzles involving the Student Six

by Coyotek4

Puzzle 17 solution

The group ponder the puzzle for some time …

“If X represents any sequence of blocks, then LXR will lead to X. Further, if X leads to Y, then note the following: KX leads to YL, QX leads to YR, DX leads to Y ‘doubled’, and CX leads to Y ‘curtailed’ … find the sequence that leads to itself.”

… and after some time had passed, little has been accomplished.

“Yona hope letters don’t appear in nightmares after this.”

Sandbar ponders the yak’s comment. “Not sure what a nightmare involving letters would look like.”

“With Headmare Twilight, the letter ‘F’ would probably keep her awake.” The group chuckles at Smolder’s insight, Rainbow Dash included.

“So, this one’s really got you stumped, huh?” Rainbow asks the students.

Ocellus continues to peruse the descriptions on the board. “Since LXR leads to X, I feel like we need some sort of X that leads to LXR. But after that, I’m stumped.”

“It looks like everything begins with LXR,” Gallus adds, “so maybe we need to focus on what comes before LXR.” He picks up a piece of chalk and writes on the board:

LYR --> Y ………… XLYR --> ???

“So ‘X’ is basically any sequence of letters that can act on a given string,” he explains. “Since the only other letters that do anything are K, Q, D, and C, ‘X’ probably has to be some sort of combination of those letters. But what do we want to get from them?”

“Well we want to get back what we start with,” Smolder adds, coming to the board to replace the question marks:

LYR --> Y ………… XLYR --> XLYR

“Wait a minute,” Silverstream says as she approaches the board. “ ‘Y’ is anything right … but it’s a fixed anything. Like, we can put it anywhere on the right, so what if …”

She erases one of the X’s and replaces with another Y:

LYR --> Y ………… XLYR --> YLYR

Yona studies the new line. “Soooo … if X is Y, then XLYR is YLYR, and we get solution, right?”

“You’re right,” Sandbar adds. “So now it’s like a construction puzzle. How do we build YLYR from just Y?”

“There’s two Y’s,” Ocellus notes, “so we’ll need a D somewhere. But not right away, because there’s an L between them.”

“So add the L first,” Smolder adds. “That turns Y into YL, and if we then double, we get YLYL.”

“And now we cut last L off,” Yona adds. “And replace with R!”

“It looks like we have all the tools we need,” Gallus concludes. “Let’s see … add the L, then double, then cut off the last L and add the R. In order, those correspond to K, D, C, and Q … but they have to be written from right to left, right?”

“Wait, left or right?” Silverstream asks.

Gallus puts a palm to his face. “Let me rephrase that last part: they have to be written from right to left, correct?”

“Oh … yeah, right … I mean, correct. That sounds right … I mean …”

“We get it!” Smolder exclaims in slight exasperation. “So from right to left, that’s QCDK. That’s our X, right?”

“And that means we should use QCDKLQCDKR,” Sandbar concludes.

“Only one way to find out,” Gallus states as he writes down a check on the board:

LQCDKR --> QCDK
KLQCDKR --> QCDKL
DKLQCDKR --> QCDKLQCDKL
CDKLQCDKR --> QCDKLQCDK_
QCDKLQCDKR --> QCDKLQCDKR

...

Rainbow Dash inspects the work shown on the board. “Wow … it looks like you did it! You guys are as good at this as Quibble was in that chamber.”

The students turn to their professor. “What chamber?” Sandbar asks.

“The marble chamber underneath the trap door in the Spectrum Catacombs in the library of the Crystal Empire where we …”

Too late, Rainbow Dash catches herself before saying any more, aware that she had said too much already. “Uh, saaay, look at the time. Guess we’re done our lessons for today!”

“You mean the friendship lessons that we seem to have stopped having ever since we started getting these puzzles?”

“Exactly,” Rainbow answers Gallus. “Gotta run!” Quickly, she turns and flies out, leaving a rainbow-colored trail in her midst.

The students look towards each other. Soon after, Smolder sighs and turns to the door.

“Where Smolder going?” Yona asks.

“To track down Rainbow Dash and tell her that Ember already shared with me her story regarding the Chalice.” Smolder proceeds out the door, leaving the others with still more questions.

“If this about Daring Do,” Yona adds, “yaks already know she exist. Daring spotted outside of Yakyakistan in past.”

“Griffinstone, too,” Gallus adds. “Not sure what the big deal is.”

Silverstream and Sandbar look at the two with wonder in their eyes. “Wait,” the latter says, “Daring Do is real? Like, a real pony?”

“Uh oh … maybe Yona say too much.”

Gallus sighs. “Just … maybe, keep it to yourself for now.” He turns to Ocellus. “So have the changelings ever been paid a visit?”

Ocellus pays no mind, her focus on the chalkboard.

“Ocellus? You OK?”

Her concentration broken, Ocellus turns to Gallus. “Huh? Oh, sorry, just that …”

“I know that look,” Gallus says. “What did you find?”

“It’s just … there’s multiple solutions to this puzzle.”

“Really?” Silverstream asks, essentially speaking for the group.

“Well, yeah. I mean, you could cut off and replace the last letter as often as you like, so the letters C and Q could be repeated in X. That means a string like QCQCDKLQCQCDKR would also lead to itself. Or QCKCDKLQCKCDKR, which adds an extra L before cutting it off for another R. You could even add and cut off a letter before doubling, so QCDKCKLQCDKCKR and QCDKCQLQCDKCQR work too.”

“We should probably tell Professor Rainbow Dash to tell her friend about that,” Sandbar says. “At least if only one solution is intended.”

“Story could ask for shortest solution,” Yona adds. “That would fix, right?”

“Good thing there’s no actual machine to punish a different answer, then,” Silverstream states.

“So far as we know,” Gallus adds. “Anyway, may as well head to our next class.”

“You think someone else is gonna ask us to solve something?” Sandbar asks.

Gallus turns to face his pony friend. “At this point, I’d be surprised not to have anyone ask us to solve something.”