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

by Coyotek4

Puzzle 13: Button, Button, Which One's 'Gray Button'?

“Fake wings?”

“Fake horns?”

Coccyx nods his head at both phrases. “All in the name of pony type anonymity. Quite a fascinating backstory, actually; by not revealing their true nature while adhering to a higher moral code, the Alicorn Society was able to bridge the gap between unicorns and pegasi … and later, earth ponies.”

“But wouldn’t that make puzzle solving, like, waaay harder?” Silverstream asks.

Coccyx chuckles. “Indeed … though before the inclusion of earth ponies and the town of Elysium, it actually led to some fascinating logical truths.”

Hearing a chorus of ‘Oh?’s, he continues: “Here’s a simple example: suppose you met one of these ‘alicorns’, and I give you the knowledge that she is not an earth pony and is not from Elysium. The alicorn states that she is, in fact, a unicorn. What could you conclude?”

The class goes silent for several seconds. Then, Sandbar speaks up:

“If she’s telling the truth, then she must be from Utopia. If she’s lying, then she’s not a unicorn, so she must be a pegasus … and she can’t be from Paradise, so she must be from Utopia.”

“Exactly,” Coccyx concludes. “An alicorn that claims to be a unicorn must be from Utopia, but could be either pony type, unicorn or pegasus. And at the same time, an alicorn that claims to be from Utopia must, necessarily, be a unicorn from either Utopia or Paradise. In short: stating that she is a particular pony type reveals her hometown, and stating her hometown reveals her pony type. I really think there’s an elegance to the symmetry, don’t you?”

Ocellus, Sandbar, Silverstream, and Yona nod in agreement, while Smolder and Gallus each shrug off the perceived beauty with a ‘meh’; the latter two cause Coccyx to laugh again. “I suppose such things are not for every creature,” he admits.

“But … symmetry gone with Elysium and earth ponies?” Yona asks.

“Indeed … without the knowledge I gave you, the previous example would have multiple possibilities. Now if she were telling the truth and was a unicorn, then of course she would still be from Utopia; but now, lying doesn’t mean she’s from Utopia: she could be a pegasus from either Utopia or Elysium, or she could be an earth pony from Utopia or Paradise. So now there’s five possibilities, and further information would be needed to reduce this number.”

“I’m afraid to ask how much more complicated the puzzles can get with these ‘alicorns’,” Smolder states.

“You don’t have to,” Coccyx replies. “Recently, I was given a particularly daunting challenge that revolved around a quintuplet of alicorns.” A sly smile spreads across his face. “Let’s see how you fare on this one.”

The class murmurs to itself in excitement, save for Gallus. “It beats the usual history lessons,” he eventually states. “I’m game.”

“Wonderful … now before we begin, I do want to remind you of an important distinction: if an alicorn, for example, states that she is a Utopian pegasus, that is different from the alicorn stating that she is from Utopia, and then stating that she is a pegasus in a separate statement.”

“What’s the difference?” Silverstream asks.

“In the former case, you know she’s lying but she could be any of the following: Utopian earth pony, Paradisian unicorn, Paradisian earth pony, Elysian unicorn, Elysian pegasus. In the latter case, she still must be lying, but now she could only be either a Paradisian unicorn, Paradisian earth pony, or Elysian unicorn.”

“So, in one case she could be a Utopian earth pony, but not the other?”

“Exactly. In the first case, all we know is that she is not the specific specified combination; she could be either half of the combination. In the second case, she cannot be either half.”

“Got it … I think,” Silverstream replies with a tinge of confusion.

“Excellent. Well then … I was introduced to five gray alicorns. I was informed that from this group of five, every pony type and every hometown was included at least once. I was also informed that no two of them had the same combination of pony type and hometown, so there couldn’t be two Utopian pegasi, for example. One of these five, I was told, went by the name ‘Gray Button’; my task was to determine the pony type and hometown of all five, as well as identify which one was, in fact, Gray Button.”

Ocellus begins calculating: “3 possibly pony types and 3 possible home towns for each alicorn, so 9 times 9 times 9 times 9 times 9 is … 59,049 possible combinations. Not accounting for which one is Gray Button.”

“But alicorns all different,” Yona corrects, “so actually 9 times 8 times 7 times 6 times 5, or 15,120.”

“Then times 5 for Gray Button,” Gallus adds, “so actually 75,600.”

“Uh, guys?” Sandbar asks. “Can we slow down with the ‘math’ part?”

“Oh … sorry, Sandbar,” Ocellus says in apology.

“I see your math skills are high as well,” Coccyx admits. “But I assure you, there’s only 1 possibility once the statements are taken into account. I’ll continue to use ‘A, B, C, D, E’ for simplicity. These were the statements made:

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.”

“And from those statements, along with the information I provided, you have enough to figure out the pony type and hometown of all five alicorns, and state which one is Gray Button.”

“So,” Coccyx concludes, “Can you figure this one out?”

Comments

[Duckgame4life]

Ok here we go
minimum 1 of each town and type, no duplicate combos
A: unicorn, Elysium, gray is from paradise
B:not earth, paradise, 3 share a type
C:earth, not Elysium, maximum 1 utopia
D: not unicorn, not paradise, gary is Pegasus
E:Pegasus, not utopia, 3 share town
A must be lieing as unicorn and Elysium contradict, so they are: Pegasus/utopia, earth/utopia or earth/paradise, either way gray isnt from paradise
If B is true they are Pegasus/paradise
If not they are earth/utopia
C cant be true, so they are Pegasus/Elysium or unicorn/Elysium and there is more than 1 utopian
If D is true they are earth/Elysium
If not they are unicorn/paradise
If E is true they are Pegasus/paradise
If not they are earth/utopia
after that im stumped

[Coyotek4]

9472999
You erred with one of the five.

[Duckgame4life]

9473281
Oh wait i see it, will edit a fix

[Coyotek4]

Perhaps this puzzle turned out harder than anticipated, as this is the first one where no one posted a solution before the 'official' solution chapter.

Rather than give away the entirety of the solution all at once, I will post a snippet of the solution chapter and see if that helps. Eventually, I will post all of it; just not right away.

[Duckgame4life]

After much thinking. Ive noticed 1 extra thing
only 1 of B and E can be true, no matter who it is, one is Pegasus/paradise and the other is earth/utopia, this means A cant be earth/utopia
Now i might be able to solve
with the fact of C lieing making it certain there is minimum 2 utopians, A must be Pegasus/utopia as no other pony has that as a possible outcome
So now we have 3 confirmed combos; A is Pegasus/utopia
B and E are Pegasus/paradise and earth/utopia in some way
For E to be true one of C&D must be a unicorn/utopia, but neither have that as a possibility, so E is lieing(earth/utopia), making B the true one(Pegasus/paradise)
this means there is a Pegasus from Elysium, and only C can fit that spot
If gray is a Pegasus he must be A or C, but we cant determine that, so D must be lieing, making gray NOT a Pegasus and D HAS to be a unicorn/paradise
With the statements A and D give, gray is NOT a Pegasus and NOT from paradise, the only pony that fits is E

In the end we have

A: utopian Pegasus
B: paradise Pegasus
C: Elysium Pegasus
D: paradise unicorn
E: utopian earth, is gray button

[Eroraf86]

I only just got around to solving this. I have not looked at the partial solution.

Possibilities:
A: eP eU pU
B: pP eU
C: pE uE
D: eE uP
E: pP eU
A and C must be false, so we know that more than one is from Utopia.
B and E form a linked pair: each must be either pP or eU, but they can't be the same, so one is true and the other is false.
The above two statements in conjunction force A to be pU.
There can't be 3 ponies from the same town, as that would either create a duplicate or force C to be true. Therefore, E is false, eU, and B is true, pP.
Since B is true, there are three ponies of the same type. This forces C to be pE.
Finally, D represents the lone unicorn, uP.
Knowing that A and D are false, Gray Button is neither a pegasus nor from Paradise, so he must be E.
The final results: A = pU, B = pP, C = pE, D = uP, E = eU = GB.

Sorry I took so long to get to this one. Life happened.