“Yer just in time, sugarcube … uh, Ah mean, Headmare Twilight.”
Twilight chuckles at Applejack’s backpedaling. “There’s no need to be so formal with me.”
“Well, in front of the students is all … Ah thought a bit more formality would be in order.” Applejack turns from Twilight to the class, who all shrug in indifference.
“I’ve never been comfortable with anypony addressing me as ‘Princess’; as it is, I’m still adjusting to ‘Headmare’. Formality can be appropriate, but not among friends.”
“Uh … we can still call you ‘Headmare’, right?”
Twilight laughs at Ocellus’s question. “Whatever makes you most comfortable,” she replies before turning back towards Applejack. “So what do you have for today’s lesson plan?”
“So, for today Ah thought it’d be interesting for the students to see first-hoof how confusin’ it can be when competin’ groups try to be uncooperative towards one another. The point, of course, is to show that honesty is always the best policy.”
“I’m with you so far,” Twilight accedes. “So how does this work.”
“Ah’ve already assigned each of the students here to one of two groups; one group is called the Squirrels, and the other group is called the Chipmunks.”
Twilight ponders the names for a moment. “I think all those friendship quests you’ve been sent on with Fluttershy are starting to have an effect.”
The two share a laugh as the class quietly observes.
“Well, Ah didn’t want to just use ‘A’ and ‘B’ for this; she may have given me suggestions,” Applejack admits. “So anyway, Ah assigned three of ‘em to be Squirrels and three of ‘em to be Chipmunks. Then Ah laid down some simple rules for the students. They can each talk to only one other at a time, and Ah included a rule what causes confusion when ponies … well, any creature gets all underhoofed.”
“And what rule would that be?”
“Any statement made toward someone from the ‘other’ group has to be a lie, while any statement made towards someone from the same group has to be honest. So if, say, Gallus and Smolder are both Squirrels, they have to be honest when talking to each other … but if Gallus is a Squirrel and Smolder is a Chipmunk …”
The other four students burst out in giggles, as Gallus and Smolder sigh in unison.
“… uh, anyway, if Gallus and Smolder are in different groups, then any statement made by one to the other has to be a lie.”
“I think I got it,” Twilight states. “So I assume everyone knows their group.”
“Yes indeed.” Applejack turns from Twilight to the students. “How ‘bout a demonstration for the Headmare, y’all?”
“What are we supposed to say?” Ocellus asks.
“Ah dunno … maybe keep things simple, like makin’ comments to one another about yer affiliation.”
Gallus sighs a second time, swallows his pride, and turns to Ocellus: “I am a Chipmunk.” He pauses for a moment. “Grampa Gruff can never know I spoke those words in that order, okay?”
“Ooo, my turn!” Yona exclaims as she, too, addresses Ocellus: “Yona is Squirrel.”
Smolder then beckons Yona to face her: “We are different types.”
Sandbar goes next, getting Smolder to face him: “You are a Squirrel.” The comment draws rolled eyes from the young dragon.
“Oh, I want to go now!” Silverstream insists as she flutters towards Gallus: “You are a Chipmunk,” she states before moving on towards Sandbar: “Either Yona is a Squirrel or Smolder is a Chipmunk.”
“Whoa now, love the enthusiasm there Silverstream, but Ocellus hasn’t even had a chance to speak.” Applejack double-checks her list and thinks about the comments made. “Ah think some of you may have forgotten the rule Ah put in place. Two of those statements wouldn’t’ve been made.”
“What do you mean?” Ocellus asks.
“What Ah mean is that one of those statements is true when it should be false, and one of those statements is false when it should be true … though the other four statements hold up. It’s alright, Ah know it’s a bit confusin’ at first … that’s why Ah always believe in honesty towards all.”
Applejack turns to Twilight, noticing the latter lost in thought. A notion quickly comes to her: “Oh shoot, Twilight … Ah never even told you which group each student was in.”
“You don’t need to,” Twilight replies with a smile.
…
“If this is going where I think this is going,” Gallus says, “can I just take my seat and observe this time? I’m still worn out from the last few puzzles.”
Ok lets do this
right away i see smolder statement to yona is a paradox, so she must be an error
And since both gallus and yona say they are opposite things to ocellus, one of them is another error
So now that ive narrowed down the errors i can solve this
Sandbar says smolder is a squirrel, if thats true then they are both squirrels, if its false than smolder is a chipmunk and sandbar is a squirrel,so sandbar MUST be squirrel
Similarly, Silverstream must chipmunk due to her first Statement to gallus
Her second statement however must be false as its to sandbar, so yona isnt squirrel and smolder isnt chipmunk
So sandbar & smolder are squirrel and yona & Silverstream are chipmunk
Now Ocellus is squirrel due to yona's statement so gallus must be chipmunk
Think I got it. Not sure if it was intended to be a guess and check, but that's how I ended up solving it. At least, I think I solved it. Another fun puzzle here.
Gallus: Squirrel
Ocellus: Chipmunk
Yona: Chipmunk
Smolder: Squirrel
Sandbar: Squirrel
Silverstream: Chipmunk
Statements:
Gallus correctly lies to Ocellus.
Yona incorrectly lies to Ocellus (should have said she is a chipmunk).
Smolder incorrectly tells truth to Yona (should have lied and said we are the same).
Sandbar correctly tells the truth to Smolder.
Silverstream correctly lies to both Gallus and Sandbar.
Found the solution. Same method and answer as AstralMouse
(Yeah I read the comments after solving the puzzle and before putting my answer in my own comments)
Another guess-and-check, but a fun challenge. I think I've finally got it.
I was a bit stuck on this one until I realized that there are not, and can not be, three truths and three lies. Once I figured out the allowable configurations (2T4F or 4T2F), the rest was straightforward. Make an assumption for Gallus, and see where that leads. Whether you get two truths or two lies, one of the pair must have been one of the erroneous statements. From there, the rest follows.
Assume Gallus is a squirrel. That gives us two lies right off the bat.
The four statements about Yona and Smolder cannot all be true, so two of them must be false. The only allowable configuration tells us that Yona lied, so Yona is a chipmunk and Smolder is a squirrel.
Now that we know Yona's and Smolder's groups, we know that Smolder's true statement should have been false. That's one of the two errors accounted for. Thus, Sandbar's statement being true reveals that Sandbar is a squirrel.
By elimination, Silverstream is a chipmunk and Ocellus is a chipmunk.
Looking at the lies, the only one made between teammates is Yona's. Yona's lie should have been true.
I'm fairly confident that this is right. Let's find out. (For anyone wondering, no, I do not read other people's guesses.)
I started off guessing-and-checking but kept finding logical rules that pared down the solution space until I figured it out without actually doing much guessing. Or...I think I figured it out. I haven't checked yet!
First off, Smolder's statement to Yona must be one of the "incorrect" (as in a lie when it should be true or true when it should be a lie) statements. If it is true then Smolder should have lied to Yona and said it was false, and if it is false then Smolder should have told the truth to Yona and said it was false.
Secondly, one of Gallus' and Yona's statements must be one of the "incorrect" statements. If you run through each possibility for Gallus (tells the truth correctly, tells the truth wrongly, tells a lie correctly, tells a lie wrongly) then Yona must be doing things wrongly if Gallus is doing them correctly or vice-versa.
Using this and Applejack's statement that there is only one "incorrect" false statement and one "incorrect" true statement, we know that every other statement than these three can be taken to be correctly stated; that is, if true then the characters are on the same team and if wrong then they're on the opposite. Therefore, Silverstream must be in Chipmunks. If Gallus is lying about being in Chipmunks, then she must be lying to him about which team he's on; but in this case she must be on the opposite team, i.e. Chipmunks. Conversely, if Gallus is telling the truth then she must be be telling the truth as well, and must be on the same team, again Chipmunks.
Next, we can determine that Smolder must be telling the truth, because Silverstream's statement is true if and only if Smolder and Yona are on the same team. Therefore, if Smolder was lying and Smolder and Yona were in fact on the same team then Silverstream would be telling the truth which would mean that Sandbar was a Chipmunk. But in this case his statement to Smolder, if false, would be a second incorrect lie, when Applejack said there was only one, and if true would be a second incorrect truth, again violating Applejack's conditions. Thus Smolder must be telling the truth, and additionally Sandbar must be a Squirrel.
We also know that Yona and Smolder are on opposite teams. We can thus deduce that Gallus and Ocellus are on opposite teams as well, since otherwise Applejack's given that each team has three ponies would be violated. So Gallus must either be lying correctly or telling the truth incorrectly. But the latter case would also violate Applejack's given that there is only one incorrect statement of each type, so we know that Gallus is lying correctly and must be a Squirrel, while Ocellus is a Chipmunk. Yona must therefore be lying incorrectly, meaning that she is a Chipmunk and Smolder is a Squirrel.
Ergo, we have the Squirrels being Smolder, Gallus, and Sandbar, and the Chipmunks being Yona, Ocellus, and Silverstream.
Lemma 1: When A states to B that A is of a type, B must be of that type.
Demonstration: Either they are of the same type and A is telling the truth, or they are of different types and A is lying.
Lemma 2: When A states to B that B is of a type, A must be of that type.
Demonstration: Likewise with Lemma 1.
Gallus’ and Yona’s statements towards Ocellus suggest that Ocellus is both a Chipmunk and a Squirrel, which is impossible. Therefore, one of them is mistaken.
Smolder’s statement is impossible. Either she and Yona are of the same type, or they are of different types and she has to lie. Therefore, Smolder is mistaken.
Then everyone else’s statements must follow the rules. Following from the lemmas, we have:
1. Sandbar is a Squirrel. (Sandbar’s statement)
2. Silverstream is a Chipmunk. (Silverstream’s first statement)
3. Yona is a Chipmunk and Smolder is a Squirrel. (Silverstream’s second statement)
Then, because Yona and Smolder are of different types, Smolder must be the one who told truth instead of lie.
Yona lied to Ocellus in her statement. This could go either way:
1) Ocellus is a Chipmunk and Yona mistakenly lied.
2) Ocellus is a Squirrel and Yona lied correctly. Then Gallus lied instead of telling the truth. He is a Squirrel.
Since there are 3 Chipmunks and 3 Squirrels, 2) is impossible, (too many Squirrels!) Then, from 1), we can infer that Gallus is the last Squirrel.
Solution:
Sandbar, Smolder, and Gallus are Squirrels. Silverstream, Yona, and Ocellus are Chipmunks.