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

by Coyotek4


Puzzle 4 solution

“Now hold on, Twi … Ah didn’t tell you what student was in each group, you know that some of those six comments made are true and some false, you know that two of those comments are wrong from what they should be … and you still think you got this all figured out?”

“Well … yeah.”

Applejack takes in Twilight’s simple reply and sighs. “If y’all don’t mind, Ah think Ah’ll just do like Gallus over there and take a seat, and let your headmare do the talkin’ for the rest of the lesson.” She proceeds to her desk and takes a seat, as the others focus in on Twilight’s solution.

“So where do you begin with this one, Headmare Twilight?” Silverstream asks.

“Well to start, let’s think about Smolder’s comment.” Twilight’s suggestion causes heads to turn Smolder’s way.

“What … I just told Yona that we were different types. I thought that was alright.”

“But here’s the strange thing,” Twilight explains: “Under the rules Applejack gave, NO ONE could utter that phrase, to ANY creature.”

She pauses for the gasps to subside, then gives further details: “Suppose one Squirrel said that to another Squirrel. Then the statement would be a lie, and any statement made between two in the same group is supposed to be true. The same would hold if a Chipmunk said the statement to another Chipmunk.”

“But what if a Squirrel told a Chipmunk that?” Sandbar asks.

“In that case, the statement would be true … but the rules state that only false statements can be made between two from different groups. So no matter the pair, that statement should never be said.”

“Oh,” Smolder utters. “I guess I got a bit confused, like professor Applejack said.”

“But there were two statements made in error,” Ocellus reminds the group. “I don’t see how the other statement could be determined … and without knowing that, I don’t know how you could continue.”

“The key here,” Twilight continues, “is to focus on two statements … specifically, the two directed at you, Ocellus.”

The changeling thinks back to the two comments made to her. “Gallus said that he was a Chipmunk, and Yona said that she was a Squirrel. It seems like either could be true OR false.”

“By themselves, either comment could be made … but as you’re about to see, it’s not possible for BOTH comments to be directed towards the same creature under any circumstance.”

Really???

“Let’s start with Gallus’s comment. If true, then he is a Chipmunk, and so is the student he’s addressing. If false, then he’s actually a Squirrel and he’s lying; that means the student he’s talking to is again a Chipmunk. So either way, the recipient of the conversation is a Chipmunk; in this case, that would be you, Ocellus.”

“Weird … a comment referring to himself ends up revealing who I am.”

“Indeed … and by the same reasoning, Yona’s comment would then force you to be a Squirrel. And that’s where the contradiction lies; you can’t be both a Chipmunk AND a Squirrel.”

“Wait,” Yona asks, “did Yona make mistake? Yona not mean to.”

“Well either you or Gallus erred, but as of now there’s no way to know who erred. All that’s known is that one of those comments is wrong … and combined with Smolder’s comment, we now know that all the other comments are within the rules. Now we can turn our attention to those comments.”

“So, my comment to Smolder was OK?”

“It was, Sandbar … and your comment actually forced YOU to be a Squirrel.”

Sandbar gapes. “Really? I said that Smolder was a Squirrel.”

“Yes, you did. And if your statement was true, that would force you to be a Squirrel so you were in the same group. But if your statement was false, then Smolder would actually be a Chipmunk; since false statements are only made by creatures from different groups, you would again have to be a Squirrel.”

“That is incredible!” Silverstream admits. “So if I say that someone else is a Squirrel, that means I’m a Squirrel … and if I say that I’m a Squirrel, that forces the other student to be a Squirrel.”

“Exactly; it’s a fun property that exists here.” Twilight catches a slight derisive laugh from Gallus. “Well, I suppose it does depend on one’s idea of ‘fun’.”

“So, all that means Sandbar has to be a Squirrel and Silverstream has to be a Chipmunk, right?” Ocellus asks.

“It does … and since Silverstream and Sandbar are in different groups, that means her comment to him has to be a lie: it is NOT the case that either Yona is a Squirrel or Smolder is a Chipmunk, so the reality is, Yona is a Chipmunk and Smolder is a Squirrel. And that takes care of four of your affiliations.”

“Remarkable,” Sandbar says in awe. “And you’re right about all four of us … but what about Gallus and Ocellus?”

“Yeah, couldn’t either of them be the last Squirrel or Chipmunk?” Smolder asks.

“There’s still one fact we haven’t used,” Twilight states. “Applejack didn’t just say that two comments were in error. She said that one statement was true that should be false, and one comment was false that should be true. So now go back to your comment.”

Smolder recollects what she said. “I said that Yona and I were different types. And now you know that I was a Squirrel and Yona was a Chipmunk … so my statement was true, and should have been false.”

“And therefore …” Twilight starts.

“… the other wrong statement was false and should have been true,” Ocellus finishes. “And that means the two students involved with one of those statements had to be in the same group.”

“But Gallus and Ocellus have to be in different groups,” Silverstream continues. “So it was the other statement that was in error.”

Yona looks to the floor for a moment. “Yona not mean to make mistake.”

“Well, how about atoning by finishing the solution?” Twilight offers, bringing a smile to the yak’s face.

“OK … so, Yona’s comment to Ocellus must be false and we must be in same group … so Ocellus must be Chipmunk.”

“And that leaves Gallus to be the last Squirrel,” Smolder adds. “And … that’s it, right?”

“That is it,” Twilight concludes. “In summary: Gallus, Smolder, and Sandbar are the Squirrels while Ocellus, Yona, and Silverstream are the Chipmunks. Yona’s comment was false and should have been true, Smolder’s comment was true and should have been false, and the other comments all followed the rules: Gallus lied, Sandbar was truthful, Silverstream lied to Gallus, and she also lied to Sandbar.”

“No matter how often you do that, Twi, Ah’ll never understand how simple you make everything sound,” Applejack states.

“Comes from a good chunk of my youth spent working on logic,” Twilight says with a tinge of remorse. “Probably too much … but I’m always happy to impart such knowledge on students such as yourselves.” She glances at the clock in the room. “Oh … class time is almost up! Sorry, Applejack; I didn’t mean to take up all your lecture time.”

“Shoot, Twilight, Ah love watchin’ you do yer thing. And hey, less work for me, to be honest,” she adds with a sly smile.”

“Less work for me too,” Gallus adds instinctively. His fellow students all turn to him and scowl. “What?”

“It’s alright,” Twilight calmly says. “I’m sure Gallus will prove to be up to the next puzzle challenge … whenever that will be.”