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

by Coyotek4

Puzzle 13 solution

A: “I am a unicorn. I come from Elysium. Gray Button comes from Paradise.”
B: “I am NOT an earth pony. I come from Paradise. Three of us are the same pony type.”
C: “I am an earth pony. I do NOT come from Elysium. At most one of us comes from Utopia.”
D: “I am NOT a unicorn. I do NOT come from Paradise. Gray Button is a pegasus.”
E: “I am a pegasus. I do NOT come from Utopia. Three of us come from the same town.”

Having written the statements on the chalkboard, along with the information that there at least one of each pony type and hometown among the five, with no two being the same for both … Coccyx waits for several minutes as the class ponder the information.

“Yona confused,” the yak admits as she looks around the room, “but Yona not alone in confusion.”

“You got that right,” Gallus adds. “Where do we even begin?”

“Where did you begin,” Ocellus asks her uncle.

“Well you can’t identify the pony type or hometown from any of the five at the onset,” Coccyx explains, “but you can at least narrow down the possibilities. Try starting there.”

“Hmmm.” Ocellus focuses on the first of the five alicorns. “Well ‘A’ must be a liar; if not, she would be an Elysian unicorn, and that contradicts what we know of them. And that means that ‘A’ must be either a pegasus or earth pony, and from either Utopia or Paradise … but not a Paradisian pegasus, since they’re truthful. But that still leaves three possibilities.”

“That’s fine,” Coccyx assures. “File away that information and move on.”

“Ooh, let me go next,” Silverstream calls out.”

“Go ahead,” Ocellus replies.

“OK, so about ‘B’ … if she’s telling the truth, then she’s from Paradise so she’d be a pegasus. If she’s lying, then she is an earth pony but not from Paradise … and not from Elysium either, ‘cause that would make her truthful. So she must be either a Paradisian pegasus or a Utopian earth pony.”

Smolder follows: “What about ‘C’? If she were telling the truth, then she’d be an earth pony who’s from either Utopia or Paradise … but in either case, she’d be a liar. So she must be a liar, and she’s not an earth pony but is from Elysium. She must be either a unicorn or pegasus.”

“Yona look at ‘D’. Uh … ‘D’ either truthful or liar. If ‘D’ truthful, then ‘D’ not unicorn and not from Paradise … so ‘D’ must be earth pony from Elysium. And if ‘D’ liar, then ‘D’ unicorn from Paradise.”

Sandbar picks up the final alicorn: “ ‘E’ could be truthful or lying, too. If she were truthful, then she would be a pegasus, and would have to be from Paradise. And if she were lying, then she would be from Utopia and not a pegasus … and not a unicorn, either. So she would be an earth pony in that case.”

“Wait a minute,” Gallus interrupts. “Didn’t we conclude that ‘B’ was either a Paradisian pegasus or a Utopian earth pony?”

Coccyx chuckles. “You did … and well done, all of you. You’ve done the dirty work that needed to be done. Now you just need to put the pieces together.”

“But how,” Ocellus asks, the others looking on with similar confused looks.

“Maybe this will help.” Coccyx picks up a piece of chalk and draws a 3x3 figure onto the board, adding letters above and to the left:

-- upe
U: - - -
P: - - -
E: - - -

“The lowercase letters represent the pony types: unicorn, pegasus, earth pony. The uppercase letters represent the towns: Utopia, Paradise, Elysium. And if you recall, no two alicorns can share both pony type and hometown. Now … who can go where?”

“Well,” Ocellus starts, “we know that ‘B’ and ‘E’ could each be either a Paradisian pegasus or a Utopian earth pony. And since they can’t be the same, that means that one of them has to be a Paradisian pegasus and the other one has to be a Utopian earth pony. But I can’t tell which is which.”

“It doesn’t matter,” Coccyx says, “you now have two spots filled, at least in some order. Now what about the others?”

Yona follows: “ ‘A’ must be Utopian pegasus or Paradisian earth pony or … wait, ‘A’ not Utopian earth pony, if ‘B’ or ‘E’ has to be.”

“What about ‘C’?” Smolder adds. “She has to be Elysian, and has to be either a unicorn or a pegasus.”

“And ‘D’ has to be either a Paradisian unicorn or an Elysian earth pony,” Sandbar adds.

“Oh, let me add this to the board,” Silverstream says as she flutters to the board. Picking up the chalk, she fills in the possibilities:

-- u p e
U: - A *
P: D * A
E: C C D

“I used a star for ‘B’ and ‘E’,” she clarifies as she returns to her seat. “So where do we go from here?”

Gallus studies the letters. “No Utopian unicorns, I see. Wait, we know that ‘A’ and ‘C’ are liars, right?”

“Good catch,” Ocellus says. “I think we should wait on ‘A’s comment until later, but what about ‘C’? She said that at most one of them came from Utopia, so in fact, at least two of them do.”

“And there aren’t any Utopian unicorns,” Smolder adds, “so there has to be both a Utopian pegasus and a Utopian earth pony. Well the ‘earth pony’ slot is either ‘B’ or ‘E’; we already knew that.”

“But now we know that ‘A’ has to be a Utopian pegasus,” Sandbar states. “That’s the only way we can get at least two Utopian ponies.”

“We got one!” Silverstream triumphantly asserts. “Now … now what?

“What about ‘B’ and ‘E’ comments?” Yona asks. “One truthful and one liar, right?”

“You’re right,” Ocellus replies, studying each of their third comments. “One of them says that three of them are the same pony type, and the other says that three of them have the same hometown. So one of those facts has to be true. Which one?”

“What can we have three of?” Smolder asks.

“We don’t have any Utopian unicorns or Paradisian earth ponies,” Gallus states. “That rules out four of the six categories. There has to be either three pegasi or three Elysians.”

“But there can’t be three Elysians,” Ocellus realizes. “ ‘C’ is the only one would could be an Elysian unicorn, and she’s also the only one who could be an Elysian pegasus. She can’t be both.”

“That just leave three pegasi,” Yona concludes, “so ‘B’ truthful and ‘E’ lie. That make ‘B’ Paradisian pegasus and ‘E’ Utopian earth pony.”

Sandbar speaks up: “If three of them are pegasi, then ‘C’ has to be an Elysian pegasus.”

“And since at least one of the five has to be a unicorn,” Smolder adds, “ ‘D’ must be a Paradisian unicorn.”

“BRAVO!” Coccyx calls out. “You got the pony types and hometowns of all five alicorns! All that’s left is to figure out which one is Gray Button.”

“There’s only two comments to work with,” Gallus says, “and both of them are lies. Gray Button isn’t from Paradise, and she isn’t a pegasus. And the only one who fits both of those statements is … ‘E’. ‘E’ must be Gray Button.”

“Wonderful,” Coccyx gushes. “Top notch; top notch! I’m starting to think there’s no puzzle the six of you can’t solve.”

Other than why we’re solving so many of these, Gallus thinks to himself.

. . .

“Uncle Coccyx?”

“Yes, Ocellus?”

“I really enjoy these puzzles … but none of them seem to have anything to do with you?”

Coccyx chuckles. “I’m sure you have concerns since you and Twilight left, but I assure you, relations between the townsfolk and the visiting changelings are very much cordial. It’s been made easier since they’ve made allowances for our presence.”

“What do you mean by ‘allowances’?” Silverstream asks.

“When we first got there, some of us may have made comments that caused … friction. But we’ve come to some agreements that those in the Alicorn Society have signed off on, and those agreements have made our presence more than tolerable; they now embrace our presence.”

“Whoa,” Yona says with surprise. “Maybe yaks take lessons from changelings.”

“Dragons, too,” Smolder adds.

“And Griffons,” Gallus concludes. “So how did you get them to embrace you so much?”

“Simple, really … our presence allowed for even greater complexity of puzzles. Let me explain …”

Comments

[Eroraf86]

Narrowing down the possibilities definitely helps a lot. What I did was write down all possibilities for each of the five, and then solve from there. It takes a bit of thought, but it's definitely doable. I'll just link my solution posted on the previous chapter: 9478553