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

by Coyotek4

Puzzle 8 solution

Smolder: “No one has a number that is a multiple of 9. I have the largest number.”
Gallus: “My number is a perfect cube. Yona’s number is a perfect square. Silverstream’s number contains a ‘3’.”
Ocellus: “Yona’s number contains a ‘4’. Smolder's number contains a ‘9’. No two of us have numbers within 7 of each other.”
Sandbar: “My number contains a ‘1’. Silverstream’s number is 21. Gallus’s number is between 16 and 24.”
Silverstream “My number is 10 less than Gallus’s number. Sandbar’s number is 27.”
Yona: “Ocellus truthful. Sandbar lies. Both Gallus’s and Silverstream’s numbers prime.”

Over 30 TRUE … unless prime, in which case FALSE.
Under 30 FALSE … unless prime, in which case TRUE.

All numbers different, odd, and under 60.

Starlight peruses the parchment for minutes. “Twilight taught me a lot of things,” she admits to the students, “but we never got around to logic puzzles. I don’t know where to begin.”

“Leave everything to us,” Gallus reassures. Starlight turns to Gallus and blankly blinks. “Long story, we’ll tell you about it later,” he adds before turning to the other students. “Everyone focus on the statements ‘you’ said; divide-and-conquer seems like the best way to start … and I think I can lead us off.”

The others refocus their thoughts to their individual statements as Gallus begins the thought process:

“I said my number was a perfect cube. Now the only perfect cubes less than 60 are 1, 8, and 27. It couldn’t be 8, because all our numbers are odd. And if it were 1 or 27, my statement would have to be false.”

“Wait,” Sandbar asks, “isn’t 1 a prime number?”

“1 neither prime nor composite,” Yona explains. “1 special, like Yona,” she adds with a smile.

“OK,” Gallus continues, “so the upshot is I could never truthfully state that my number is a perfect cube. That means all my statements are false, so Yona’s number isn’t a perfect square and Silverstream’s number doesn’t contain a ‘3’.”

“Wow … I’m impressed,” Starlight states. “You say you’ve been doing this a lot?”

“More than I’d like to admit,” he concedes. “Anyone got anything else?”

“I … I think I do,” Sandbar says. “Based on what you just said.”

“What do you got?” Ocellus asks.

“Well … I said that Silverstream’s number was 21 and Gallus’s number was between 16 and 24. If both of those were true, then his number would have to be 17, 19, or 23 … but all of those are prime, which would mean Gallus’s statements would be true. So I must have lied, too. And my number doesn’t contain a ‘1’.”

“Oh, oh,” Yona interrupts, “that mean Yona truthful when Yona say Sandbar lied. So Yona truthful, and that mean Ocellus truthful and both Gallus and Silverstream have prime numbers!”

“And that means my statements are true, too,” Ocellus adds. “So Yona’s number contains a ‘4’, Smolder’s number contains a ‘9’, and all our numbers are 8 or more apart from each other.”

“Remarkable,” Starlight says in awe. “So you already know the veracity of four of your sets of statements.”

“We’re very well-versed … city,” Silverstream says. “But do we know any actual numbers yet?”

Yona ponders Ocellus’s statement regarding her own number: “Ocellus truthful, so Yona have number with ‘4’ in it … but 4 can’t be last digit, because all numbers odd. So Yona have 40-something.”

“But not a prime number,” Smolder adds, “or else you’d be lying. So … that means it’s not 41, 43, or 47.”

“It’s not 49, either,” Gallus adds, “because that would mean Yona’s number is a perfect square, making one of my statements true. So that only leaves—”

“YONA HAVE 45!!!” the young yak enthusiastically concludes.

“Awesome,” Sandbar says. “One down, five to go. So … now what?”

Smolder comments on her own statements: “45 is a multiple of 9 … so I guess I’m a liar, too.”

“I’d say you’re in good company,” Gallus responds with a grin. Smolder remains unfazed. “What, you don’t get? … whatever. As you were saying?”

“Yeah, well … since I’m a liar, I don’t have the largest number. Now, my number has a 9 in it, so the choices are 9, 19, 29, 39, 49, and 59. And it sure isn’t 59, since that’s the largest number possible.”

“It’s also not 39 or 49,” Ocellus adds, “since you’d be telling the truth if you have those numbers.”

“Same with 19 and 29,” Sandbar adds as well.

“So I guess I got 9,” Smolder concludes. “There’s our second number. Now what?”

The group studies the parchment for several minutes. “Now it gets difficult,” Gallus concludes.

“Let’s go back to your number,” Ocellus suggests. “What could it be?”

“A lot of things, right?”

“Not really. I mean, we know it’s a prime number, and that mean it’s greater than 30. It can’t be within 7 of 45, so … it can only be 31, 37, 53, or 59.”

“Ooo, now I know something!” Silverstream excitedly asserts. “If my statements were true, then my number would be 10 less than Gallus’s number. That would make my number either 21, 27, 43, or 49.”

“But Silverstream have prime number,” Yona adds. “So number would be 43.”

“And that would make Silverstream a liar,” Sandbar adds as well, “so Silverstream can’t be truthful, and her number isn’t 10 less than Gallus’s number.”

“But her number is still prime,” Smolder follows, “so it’s greater than 30 just like Gallus’s. And, her number doesn’t contain a 3. So what does that leave?”

Gallus runs through possibilities: “Greater than 30, doesn’t have a 3 in it, so greater than 40 … not within 7 of 45, so at least 53 … but that contains a 3, so …”

“Oh, I have 59!” Silverstream finishes.

“And that means my number isn’t 59 or 53,” Gallus adds, “so it has to be either 31 or 37.”

“And everyone else’s number has to be less than 30,” Ocellus states.

“Let’s go back to my number,” Sandbar suggests. “It has to be less than 30, so it’s can’t be prime. It’s not 1, it can’t be within 7 of 9, it can’t be 17 or 19, or 21 or 23. It has to be 25 or 27 … but It’s not 27 either, because of Silverstream’s statement. I have to have 25!”

“And that means I can’t have 31,” Gallus notes, “so I have 37.”

“So what’s left for my number? Can’t be within 7 of either 9 or 25, can’t be 1 … my number can only be 17.”

“That all numbers,” Yona says with finality: “Smolder have 9, Ocellus 17, Sandbar 25, Gallus 37, Yona 45, and Silverstream 59.”

…

Silence pervades for several seconds.

“Did we … just solve that puzzle?” Silverstream asks.

“YOU MOST CERTAINLY DID, YOU MEDDLING KIDS!!!”

The students all turn to the entrance and eye Discord nonchalantly standing there, eyeing the group right back.

“Technically I win the challenge, since your counselor did nothing more than watch and maybe learn. Still … the way you six demonstrated finding reasoning amidst the chaos I put into that puzzle speaks volumes:”

He snaps his fingers, and a number of tomes fly off the bookshelves, ‘flapping their covers’ as they speak accolades about the students:

“Well done, chaps.”

“Jolly good show.”

“Much better than some unicorns!”

Starlight’s eyes glare as Discord snaps his fingers a second time. The books immediately return to their places along the bookshelves.

“When indoors, it’s always proper to have volume control,” he points out. “In any event, I’m late for work, so I’ll let you be … for now.”

“WHAT???” Starlight shouts. “Since when do you have a job?”

“Since there seems to be little in the way of educational job openings, I thought I’d offer my services elsewhere to pass the time.”

“So … exactly what do you do?” Ocellus asks.

“Air-traffic control. I’ll be monitoring the hot-air balloons for the foreseeable future. Anyway, well done to the six of you … and considering yourself lucky,” he adds to Starlight before disappearing with a final *POP*.

The puzzle now behind them, the students take in one last look around the water-logged office. “You want us to get a mop or something?” Silverstream asks.

“You’ve done so much for me already,” Starlight insists. “I am thoroughly impressed what you were able to accomplish this afternoon.”

“Because we solve puzzle?” Yona asks.

“Because you got rid of Discord,” Starlight corrects. “Hopefully, now we can return to a sense of normalcy around here.”

“Maybe,” Smolder says, “but if I’m taking a break near the fountain outside, I’ll still watch for falling balloons.”

Author's Note:

Bonus points if you figured out my 'Shout-Out' at the end.

Comments

[son_of_heaven176]

OK, it's been a while, so...what was the shout-out?

And by the way, I got through most of the truth values, but didn't take the time to think through all of the implications. Well done spelling out the math. (If I wasn't doing this near midnight, I might have gotten the puzzle!)

[Coyotek4]

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