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

by Coyotek4

Puzzle 18 solution

Gallus and the others turn to Yona, with Gallus in particular observing the young yak through skeptical eyes. “Really,” he deadpans.

“Uh huh.”

“You know all six cards, despite not seeing any of them.”

“Yaks good at math.”

“You get all these cards right, and I’ll spend next Yikslirbirtfest with you in Yakyakistan.”

Yona’s eyes enlarge at the possibility.

“Ah ah ah … I want proof that you know these, not some lucky guesses.”

“Griffon on!” Yona declares.

Having pondered her own card during the conversation, Ocellus looks up and turns to Gallus. “You may want to bring a scarf.”

“And no help!” Gallus adds. “All right … what’s your reasoning.”

“OK, well … first, nocreature have ‘1’, or somecreature would have played immediately.”

The fivesome turn to each other and nod. “Can’t argue that,” Smolder comments.

“So now, three cards multiply to 360, so for example, cards could be 2, 3, and 60, but then Ocellus card 65 and then she know three cards since no other group of three cards add to 65 and multiply to 360. Instead, Ocellus must have card where two groups of three cards exist that add to Ocellus card and multiply to 360.”

“She’s right,” Ocellus adds, careful not to reveal her card. “I found two possibilities … but is my card the only card that leads to multiple possibilities?”

“No … Yona do math in head and figure out three possible sums: 23, 25, and 29. Hang on, Yona write down.”

The young yak grabs a pencil in her mouth and jots down several numbers:

4+9+10=23 and 5+6+12=23
3+10+12=25 and 4+6+15=25
2+12+15=29 and 3+6+20=29

Yona spits out the pencil. “So now Yona know Ocellus have either 23, 25, or 29; all other sets of three numbers that multiply to 360 have unique total when added.”

Gallus blinks in astonishment. “You … figured all that out. In your head.”

Yona shrugs. “Like Yona say, yaks good at math.”

“You sure are,” Smolder admits. “But now what?”

“Now Yona remember Gallus comment, that Ocellus know everything if she know either card of Gallus. But that can only be case if Gallus cards are 6 and 12.”

“Why is that?” Sandbar asks.

“Because 6 and 12 are only numbers that appear in all three rows. So if Ocellus know Gallus has 4, and Ocellus have 23 or 25, then Ocellus know other cards, but not if Ocellus have 29. Or if Ocellus know Gallus has 3; then Ocellus know if she have 25 or 29, but not if she have 23.”

“Huh … Guess I’ll need that scarf after all,” Gallus admits, revealing his two cards to the others: 6 and 12.

“And since only one combination have neither 6 nor 12, Ocellus must have 23 and other cards must be 4, 9, and 10.”

Ocellus reveals her card, the 23, and the other three show their cards, confirming the solution Yona pointed out of 4, 9, and 10. “You never cease to amaze me,” Smolder admits.

“Wait a minute,” Silverstream says, an idea popping into her head as she turns to Gallus. “The only way for this puzzle to make sense is for you to know all this when you made your comment … you did all that math in your head, too?”

Gallus shrugs, his eyes darting around the room. “Well … yeah. I’m good at math; I just don’t like to brag about it. Remember when Trixie gave that guest lecture about her first visit to Ponyville?”

Sandbar turns to Ocellus. “And when you commented that Gallus might need a scarf … you figured this all out, too, didn’t you?”

Ocellus blushes. “I just like puzzles,” she meekly replies.

Smolder’s gaze turns from Gallus to Ocellus to Yona. “I suppose when we all graduate, the three of you could turn around and open a school for mathematical logic.”

“Don’t go selling yourself short,” Gallus replies. “I think we’ve all shown a lot over the past several weeks.”

“Mostly the rest of you,” Sandbar says with a tinge of regret. “I feel like I’ve been more of a follower than a leader during most of these.”

Ocellus ponders Sandbar’s comment. “If you feel that strongly about it, there’s still some other ordering tests that the rest of us took that you haven’t heard about.” Her words elicit a groan from the others. “What?”

“You want to dredge up those meaningless tests?” Smolder asks. “Those things were completely pointless.”

“But not useless,” Ocellus corrects. “I think I could create an interesting puzzle based on some of those.”

Author's Note:

Ocellus alludes to this previous one-off logic puzzle, which spawned this collection:

A Question of Rankings

Comments

[Eroraf86]

9549856
Ah, good catch.

[Another Nopony]

And solved. The next step is to focus on Gallus's statement. Since revealing either of his cards would tell Ocellus what the three cards on the table were, that means that we need to find two numbers which eliminate one possibility for each sum. The only two numbers which fit this criteria are 6 and 12. Of the six possible combinations, there is only one which contains neither of these numbers: 4, 9 and 10 adding to 23.