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

by Coyotek4

Puzzle 12: Silent Pegasi

“So Ocellus,” Coccyx starts, “care to introduce me to your fellow classmates.”

“Oh … well,” Ocellus replies, “these are Sandbar, Smolder, Silverstream, Yona, and Gallus.” Each student nods upon hearing his or her name. “And … those are my classmates … and my friends.”

“It’s a pleasure to meet all of you,” Coccyx states. “Do any of you have any questions about me?”

“From what Ocellus has told us,” Gallus says, “you’ve been spending time in some pony village that served as inspiration for some logic puzzles. That seems … odd.”

Coccyx laughs. “Yes, the twin towns of Utopia and Paradise … now the triple towns, including Elysium. It’s not as foreboding for travelers as it used to be, at least from what I’ve heard of their history. However, they still very much embrace what made their towns special.”

“Oh? What would that be?” Silverstream asks.

“It would take too long to delve into what I know of their history, so I’ll keep it simple. Utopia and Paradise were founded by unicorns and pegasi, and over the years a quirk developed regarding the inhabitants of those towns: in Utopia, all the unicorns tell the truth and all the pegasi lie; in Paradise, it’s the opposite: all the unicorns lie and all the pegasi tell the truth.”

“That’s … really weird,” Sandbar admits.

“I’m only giving a summary, but I’ve been told that the twin towns were the basis of a number of logic puzzles. As a simple example: suppose you came across two unicorns, either of which could come from either town. One unicorn states that they both come from Paradise. Could you tell which town each come from?”

The group ponder the question for a few minutes. Soon after, Yona responds:

“First unicorn say both from Paradise … but you say all unicorns from Paradise lie, so it can’t be that both unicorns from Paradise.”

“On the other hoof,” Sandbar adds, “the first unicorn lied, which means she is from Paradise.”

“But since the statement was false,” Silverstream continues, “they can’t both be from Paradise … so the second unicorn must be from Utopia!”

“Very impressive,” Coccyx admits to the class. “And a very quick response, no less,”

“We’ve had some practice as of late,” Gallus states.

“But what about earth ponies?” Smolder asks.

“Ah, that’s where more recent history comes into play. Initially, earth ponies were not even part of either town … they were left behind and forced to form a third town, away from the others; that town, they dubbed ‘Elysium’.”

“Wait,” Yona says, “so ponies not always friends?”

“I think that was the basis of Hearthswarming,” Ocellus replies. “Long ago, there was animosity among the three pony types.” Sandbar nods in agreement.

“And from what I know,” Coccyx continues, “that animosity existed in some form when these towns were formed. Now, I can tell you that things aren’t that way anymore. All three towns now sport all pony types … but in a nod to the old traditions that made their towns unique, they still adopt their truthfulness regarding place of residence to the old ways.”

Gallus thinks for a moment. “So what that means is …”

“In terms of where each come from, the only truth-tellers are Utopian unicorns, Paradisian pegasi, and Elysian earth ponies. The unicorns from either Paradise or Elysium lie, as do the pegasi from Utopia or Elysium, and the earth ponies from either Utopia or Paradise.”

“That’s … even weirder,” Silverstream states.

“I know … it’s weird to me, too. But it does make for interesting puzzles; for example, it would render the previous puzzle as having multiple solutions: now the first pony could be from Elysium and the second pony could be from any of the three towns. Or for that matter, the first pony could still be from Paradise, but now the second pony could be from Elysium.”

The class silently takes in the explanation.

“Here’s another, simpler example: if a unicorn from one of the three town stated that he was from Paradise, what could you conclude?”

After a brief pause, Smolder offers a quick response: “The unicorn can’t be from Paradise, since then he’d be telling the truth. But he can’t be from Utopia either, since all Utopian unicorns are truthful. So that means he must be from Elysium.”

“Very good,” Coccyx states. “Maybe you guys need a bigger challenge,” he adds with a sly grin.

“What sort of challenge?” Sandbar asks.

“Recently, I and a number of fellow changelings have acclimated ourselves to the customs of the triple towns. The elder statesponies of the towns insist that we changelings understand and abide by these customs, and as such have created a number of challenges to keep us on our hooftips. One such challenge was presented to me just this past week; I’m curious if you are able to solve this puzzle.”

Gallus snorts. “Friendship is logic, I guess. Go ahead.”

Coccyx chuckles. “OK then. As a test, I was introduced to six ponies: two unicorns, two pegasi, and two earth ponies. I was told in advance that of each pair, one would tell the truth and one would lie, in accordance with the rules stated beforehoof. For simplicity’s sake, I’ll refer to the unicorns as A and B, the pegasi as C and D, and the earth ponies as E and F. They made the following statements:

A said “None of the others have the same hometown as me.”
B said “F and I have the same hometown.”
E said “C is from Utopia.”
F said “D and E have different hometowns.”
…

“What about the pegasi?” Ocellus asks.

“For some reason neither one said anything,” Coccyx explains. ‘Pega-shy’ was more like it, I guess.”

“Well it’s not like we don’t know a shy pegasus,” Gallus admits. “So did that invalidate the test?”

Coccyx smiles. “On the contrary, there’s enough information to figure out all six ponies’ hometowns.”

Author's Note:

So … what are the hometowns of each of A, B, C, D, E, and F?

Comments

[Duckgame4life]

Now i remember these puzzles
A says nobody shares the same hometown as A,so either A is utopian or A and at least 1 other share a hometown
B says he and F are the same town, for that to be true both are from utopia, witch makes F a liar
If F lies, E is truthful, making them Elysium and C utopia, and D from paradise,but that makes F truthful
If F is true, he is Elysium, making E a liar
C is therefore not utopian
B is also lieing so they are from paradise
A is therefore true and from utopia, and nobody else can be
That makes E is from paradise
In order for F to be true and E false, D is Elysium and C is paradise
So the answer is:
A:unicorn from utopia
B:unicorn from paradise
C:Pegasus from paradise
D:Pegasus from Elysium
E:earth from paradise
F:earth from Elysium

[Eroraf86]

This is a fun one.
Start with the more restrictive assumption that A is true, ergo A is from Utopia.
E cannot be true, so E is from Paradise and F is from Elysium.
Since B is false, B is from Paradise.
F is true, so D is from Elysium. Therefore, C is from Paradise.
I might have to go back and read those two stories. If they have logic puzzles like this, they might be good fun.