//------------------------------//
// Puzzle 14 solution 
// Story: School of Logic: a Collection of Puzzles involving the Student Six 
// by Coyotek4 
//------------------------------//



A: C is a pegasus.
B: A is from Utopia.
C: D is a changeling. A is a unicorn.
D: Only one of us is a liar. B is from Paradise.




A recruit who took the test before you was then told which alicorn was from Elysium, but that recruit still did not have enough information to solve the puzzle.

Another recruit who took the test before you, who knew nothing of the first recruit, was then told which alicorn was the earth pony; that recruit also has insufficient information to solve the puzzle.



“Where to begin, where to start, what to do, what to think, …” Silverstream rambles to herself for several seconds, before Ocellus turns to her.

“Maybe we should start with ‘D’s first statement. What if it’s true?”

Gallus responds: “If it is, then 3 of them are truthful and 1 is a liar. Could all 3 truthful alicorns be ponies?”

“That would force them to be a Utopian unicorn, Paradisian pegasus, and Elysian earth pony,” Sandbar replies. “Also, the changeling would be a liar.”

“That not work,” Yona counters. “That make ‘A’ and ‘C’ both liars.”

“Why so?”

Yona answers Ocellus’s query: “ ‘C’ say ‘D’ changeling, so if ‘D’ truthful, then ‘C’ must be liar. And ‘D’ say ‘B’ from Paradise, so ‘B’ must be pegasus, make ‘A’ liar.”

“She’s right,” Smolder states. “So the 3 truthful ones can’t all be ponies. It would have to be 2 truthful ponies and a truthful changeling.”

“That can’t work either,” Silverstream says. “The changeling isn’t ‘balancing’ in that case.”

Ocellus adds: “Besides, given what we know there can’t be exactly 2 truthful ponies; the third would have to be truthful, assuming there’s one of each pony type and one from each hometown.”

Gallus concludes: “All of which means ‘D’ is lying, as well as at least one other. And I guess that means ‘B’ isn’t from Paradise … what now?”

“What about ‘C’?” Smolder asks. “Could ‘C’ be truthful?”

“If so,” Ocellus answers, “that makes ‘D’ a lying changeling. And that would mean either 2 or 3 truthful ponies. It still can’t be exactly 2, and if it were 3, ‘D’ would be truthful; we already ruled that out.”

“So ‘C’ is a liar, too,” Smolder concludes. “And ‘A’ isn’t a unicorn. Anything else?”

“Well at least one of the others is truthful,” Sandbar states. “Could both of them be?”

“Can’t be,” Yona responds. “If both truthful, then ‘A’ truthful Utopian, would have to be unicorn. That not work if ‘C’ liar.”

“All right,” Ocellus says. “So far we know that exactly one of them is truthful, and it’s either ‘A’ or ‘B’.”

…

“We might be stuck now,” Gallus laments.

“You give yourself too little credit,” Coccyx insists. “Well done, all of you! You’ve come farther than most. But now I suggest you start figuring out possible solutions, and then use the additional information I provided.”

“Possible solutions, eh?” Gallus ponders the suggestion. “OK then, we know that either A or B is the only truthful one, and whichever one she is is also the changeling. So I guess we just ‘pick a path’.”

“ ‘A’ or ‘B’?” Ocellus asks the group.

“May as well start with ‘A’,” Smolder recommends. “If ‘A’ is the changeling, then ‘C’ is the pegasus … but could be from either Utopia or Elysium.”

“If ‘C’ is from Utopia,” Sandbar adds, “then ‘B’ has to be from Elysium; she can’t be from Paradise, since ‘D’ lied.”

“That make ‘D’ from Paradise,” Yona continues. “And if ‘C’ Utopian pegasus, then … ‘B’ must be Elysian unicorn and ‘D’ Paradisian earth pony.”

“Maybe we should put this on the board,” Silverstream suggests as she flutters over to the front of the class and picks up a piece of chalk. “So, Case 1: ‘A’ changeling, ‘B’ Elysian unicorn, ‘C’ Utopian pegasus, ‘D’ Paradisian earth pony”

“OK … but what if ‘C’ was from Elysium instead?” Ocellus asks.

“ ‘B’ still can’t be from Paradise,” Gallus points out, “so she has to be from Utopia. And ‘D’ would have to be from Paradise.”

“And ‘B’ can’t be the unicorn then,” Smolder adds, “so she’s the earth pony and ‘D’ is the unicorn.”

“Got it,” Silverstream states as she writes down the next possibility: “Case 2: ‘A’ changeling, ‘B’ Utopian earth pony, ‘C’ Elysian pegasus, ‘D’ Paradisian unicorn”

“That’s all the possibilities with ‘A’ as the changeling,” Sandbar notes. “What about ‘B’?”

“If ‘B’ changeling,” Yona starts, “ ‘A’ from Utopia but still not unicorn. Either pegasus or earth pony.”

“If we assume ‘pegasus’,” Ocellus says, “then ‘C’ and ‘D’ have to be a Paradisian earth pony and an Elysian unicorn, in some order.” She ponders the original statements for several seconds. “But there’s no way of knowing which is which.”

“Looks like Silverstream’s got double-duty, then,” Gallus states. “Can you add those cases?”

“I’m on it,” Silverstream assures: “Case 3: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Paradisian earth pony, ‘D’ Elysian unicorn … and Case 4: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Elysian unicorn, ‘D’ Paradisian earth pony”

“Only one path left to take,” Ocellus states, “If ‘A’ is an earth pony.”

“ ‘C’ can’t be a pegasus in this case,” Smolder adds, “so she’s the unicorn and ‘D’ is the pegasus.”

“Then ‘D’ not from Paradise,” Yona concludes, “so ‘D’ from Elysium and ‘C’ from Paradise.”

“Last one,” Silverstream echoes as she writes the fifth possibility down: “Case 5: ‘A’ Utopian earth pony, ‘B’ changeling, ‘C’ Paradisian unicorn, ‘D’ Elysian pegasus”

Silverstream flutters back to her seat, and the students take in the five possibilities:



Case 1: ‘A’ changeling, ‘B’ Elysian unicorn, ‘C’ Utopian pegasus, ‘D’ Paradisian earth pony
Case 2: ‘A’ changeling, ‘B’ Utopian earth pony, ‘C’ Elysian pegasus, ‘D’ Paradisian unicorn
Case 3: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Paradisian earth pony, ‘D’ Elysian unicorn
Case 4: ‘A’ Utopian pegasus, ‘B’ changeling, ‘C’ Elysian unicorn, ‘D’ Paradisian earth pony
Case 5: ‘A’ Utopian earth pony, ‘B’ changeling, ‘C’ Paradisian unicorn, ‘D’ Elysian pegasus



“We’ve used everything,” Smolder states.

“No we haven’t,” Gallus corrects, the realization dawning on him. “We were told that two recruits before us were given additional information, and that information wasn’t enough in either case.”

“So what?” Yona asks.

“So consider which alicorn could be from Elysium,” Gallus responds. “In order, the cases state that the Elysian is either B, C, D, C, or D. But it can’t be ‘B’ … or that first recruit would have had enough information!”

“But if that recruit was told that either ‘C’ or ‘D’ was the Elysian,” Ocellus adds, “that wouldn’t be enough.  We can rule out Case 1!”

“That still leaves 4 cases,” Sandbar points out.

“But now focus on who the earth pony could be,” Smolder states, the final piece coming into place for her as well. “Going in order again, the earth pony is either D, B, C, D, or A … the only one that doesn’t immediately give away the answer is if ‘D’ is the earth pony.”

“But didn’t we rule out Case 1?” Yona asks.

“But that’s just it,” Ocellus answers: “We know it’s not Case 1, but the second recruit doesn’t know that! If Case 4 is the actual solution, then the first recruit being told that ‘C’ was the Elysian would still allow the possibility that it’s Case 2 … and the second recruit being told that ‘D’ was the earth pony would still allow the possibility that it’s Case 1.”

“So … we’re done, right?” Silverstream asks. “It’s Case 4, isn’t it?”

“It is!” Coccyx admits: “Indeed, the final solution is that ‘A’ is a Utopian pegasus, ‘B’ is a changeling, ‘C’ is an Elysian unicorn, and ‘D’ is a Paradisian earth pony. EXCEPTIONAL WORK, EVERYCREATURE!”

“Sounds like I missed some good work.”

Coccyx and the others all turn to the doorway, observing Twilight as she observes them.

“What are you doing here?” Ocellus asks.

“More like ‘what are you all still doing here?’ … you’ve been here for an hour and a half.”

The group turn to the clock on the wall, noting that indeed, ninety minutes had elapsed since Coccyx entered the room.

“Wow,” Gallus says, “I don’t think we’ve ever had a class go by so fast.”

“I guess time is relative,” Twilight replies, “but I’m afraid you do have other classes to get to.” She turns to Coccyx: “Thank you again for taking time out of your busy schedule to visit our school.”

“The pleasure was all mine,” Coccyx insists. “Your students work exceedingly well together … something I suppose I should expect from true friends.”

Silverstream, Yona, Sandbar, and Ocellus smile and blush, as Smolder and Gallus simply sigh and smile.

“Well, I should be heading back,” Coccyx tells the class. “It was great seeing you again, Ocellus … and it was a real pleasure to meet your classmates.”

“Take care, uncle Coccyx … hope everything continues to go well at the twin … er, the triple towns.”

Coccyx chuckles. “Hey, maybe a field trip is in order at some point.” He again turns to Twilight. “What do you say?”

“There’s a lot on our schedule,” Twilight sadly replies … “but we’ll see. Maybe next semester.”

“Well then … until next time ...” Giving one final bow, Coccyx turns and flies out of the room.


Twilight scans the chalkboard. “Wow … sorry to have missed whatever puzzle this came from.”

“Don’t sweat it,” Gallus says, “these days, the next one’s probably just around the—”

*OOF*

“… corner?”

The class gather around Twilight and get her back on her hooves.”

“Oh dear, I am so sorry,” Rarity pleads. “I was just looking for … oh, there you all are!”

“Our apologies, Miss Rarity,” Sandbar says. “We kinda went over our time with the last class.”

“Oh, that’s quite all right,” Rarity insists. “But come; there’s fashion to be done!”

Satisfied that Twilight is no worse for wear, Rarity heads back out the doorway, with Twilight and the students following in tow …