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

by Coyotek4

Puzzle 11 solution

Author's Note:

The original version of the puzzle contained incorrect information, as mentioned in the 'Comments' section for that chapter. Those who were stuck the first time may wish to re-read that chapter before continuing with the solution below.

“If those were just guesses,” Gallus states, “I need to introduce you to the Griffonstone Lottery.”

“Or maybe trip to Las Pegasus,” Yona adds.

“They weren’t pure guesses,” Ocellus insists. “I knew that at least one of those numbers had to be; the luck was that your scores differed by more than one, which forced you to have those scores.”

“But how could you know?” Gallus reiterates. “Your head was buried in that tome the whole time; you never saw us nod or shake our heads.”

“Yeah, so how you tell what we answer?” Yona follows.

“It’s all about inference,” Ocellus says as she begins her explanation. “The key is that each of you thought you knew the other’s score. So, start with Yona’s question to Gallus. Yona, you asked whether or not Gallus’s score was a multiple of six.”

Yona nods.

“Well suppose Gallus said ‘no’.”

“Gallus not say anything,” Yona corrects. “Gallus just nod head.”

Ocellus sighs. “You know what I … never mind. Suppose Gallus shook his head to mean ‘no’. Then how could you possibly know his score? I mean, your score could be 8, or 10, or 13, and you wouldn’t be able to figure much out about Gallus’s score. From your view, he could have a score of 1 or 2 or 3, right?”

Yona ponders Ocellus’s statement. “Yona guess that make sense.”

“Now it could be possible for you to know Gallus’s score … but only if your own score was 2, in which case you would know his score was 1. Anything higher, and you couldn’t know if his score was 1, 2, or possibly something else. And if your score was 2, Gallus would have a score of 1 and he would NEVER know your score after his one question.”

“OK, OK, you got me,” Gallus admits. “Yes, I nodded to Yona’s question: I did claim that my score was a multiple of six.”

Ocellus smiles. “All right, so now I figured out what Yona’s score could be.”

Yona interrupts: “How? Yona not say whether number triangular or not.”

“True, but again, you claimed to know Gallus’s score. So suppose your score was 13 or higher. How could you know whether Gallus got a score of either 6 or 12?”

Yona struggles to respond.

“Exactly … you wouldn’t know. And that means your score has to be between 7 and 12, and you thought Gallus’s score was 6.”

Yona stares at Ocellus for a moment, then flips over a scrap of paper near her, revealing a guess of ‘6’ for Gallus’s score. “Yona impressed.”

“Me too,” Gallus concedes. “But you narrowed it down further, didn’t you.”

“Yes, because now I can focus on your question to Yona. You asked if her number was triangular; if she shook her head to say ‘no’, then you would have no way of knowing whether her score was 20 or 19, or possibly other values, unless your own score was 19 … and In that case, Yona would have a score of 20, which we now know cannot be.”

“Yeah, Yona did nod her head, just like I did with her question to me.”

“So now you’re thinking ‘Yona’s score is a triangular number.’ … what could that number be? The triangular numbers less than or equal to twenty are 1, 3, 6, 10, and 15.”

“You know triangle numbers?” Yona asks in surprise.

“I do … I know a lot about numbers,” Ocellus admits with a tinge of embarrassment.

“I suppose I really shouldn’t be surprised,” Gallus states.

“OK,” Ocellus continues, “so how can you think to know her score? If your own score was 9 or less, you couldn’t know whether her score was either 10 or 15 … so your score has to be between 10 and 14, and you must have thought Yona’s score was 15.”

Gallus picks up the scrap containing his guess and shows it to Ocellus, confirming her belief. “Right again, you little detective.”

Ocellus smiles at the comment. “So to sum up: Yona must have a score between 7 and 12, thinking Gallus scored 6, and Gallus must have a score between 10 and 14, thinking Yona scored 15. But Gallus’s score is less than Yona’s, so the only possible combinations are 10 and 11, 10 and 12, or 11 and 12. No matter what, either Gallus scored 10 or Yona scored 12, so that’s what I guessed … and, well … I got lucky with both, I guess.”

...

“Wow … it’s like you’re in our heads,” Gallus states.

“Yona prefer head privacy … but that was incredible.”

“Thank you both,” Ocellus says. “I’ll just, uh, get back to my book, then.”

As she turns to return to the couch, Gallus catches her with a closing question: “So are all changelings as smart as you with logic?”

Ocellus chuckles. “Funny you should ask … Headmare Twilight pulled me aside after classes today to tell me to expect a visitor tomorrow. That’s kinda the reason why I’m rereading this,” she adds as she indicates the tome on the couch.”

“Ooo, Ocellus have family visit? Can we meet with Ocellus family?”

“Actually, Headmare Twilight says he’ll be a special guest lecturer for us tomorrow.”

“Really?” Gallus asks. “Who’s coming to speak to us?”

“My uncle Coccyx.”

Comments

[Eroraf86]

Coccyx? Bit of an odd name for an insect-themed species. Still, this should be interesting.

[Duckgame4life]

YA CROSS OVER

[Coyotek4]

9453635
Uncle Coccyx is a character I created for a previous logic-puzzle Fic. I plan on posting puzzles similar to what I used in that one for the next chapter.

If you're curious: Twilight's New Logic Puzzle Adventure
(which itself is a sequel to this: Twilight's Logic Puzzle Adventure)

[GeneralOs]

The problem I have with this is that gallus announces he knows Yona's score. This gives Yona extra information. For example, if Gallus knew Yona's score was triangular and he knew it immediately this means he had to have a score above 10 the 2nd largest triangular number. He announces this which tells Yona he has a score of 12 since he answered yes to the multiple of 6 questions and Yona has to have the higher score so it can't be 18.

So I solved it to the point where Gallus thought Yona had 15 and Yona thought Gallus had 12.
I couldn't go any further from that point since there were like 5 possible numbers for Yona and 3 for Gallus.

To clear this up have them both say it at the same time on the question page EG "I know your score," Yona and Gallus announced.

[Coyotek4]

9453817
I'm starting to think the 'Acacia Avenue' puzzle I based this off of didn't take this into account, either.

I'll tweak the puzzle as you suggested. Then I'll probably avoid trying to create any more of these 'inference' puzzles.

[GeneralOs]

9453848
Thank you