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

by Coyotek4

Puzzle 3 solution

“Seriously, Ocellus, do changelings have the power to read minds as well as transform into any creature?”

“That would be cool,” Yona admits.

Ocellus puts down her book and trots over to Gallus and Yona. “It’s nothing so out-of-the-ordinary. I just took the information I had and deduced what your responses must have been.

“But you didn’t see any of my responses,” Gallus states incredulously. “I mean, with 4 questions, each one of which could be answered with ‘yes’ or ‘no’, that makes for …”

“… 2 to the fourth, or 16 possibilities,” Yona finishes.

“… OK, I won’t doubt your math anymore.” Gallus turns from the yak to the changeling. “So yeah, 16 possibilities. How could you know the answers I gave?”

Ocellus proceeds with her explanation: “The key is the fact that Yona thought all your responses were the opposite of what you intended. That meant there had to be a series of responses so that both the actual AND negated responses led to only a hoofful of possibilities.”

Yona and Gallus both look blankly at Ocellus, who chuckles slightly before continuing.

“Take Yona’s first two questions: whether or not the number was a perfect square, and whether or not it was a perfect cube. Now if Yona saw what she believed to be two ‘yes’ answers, she would have immediately thought the number was 729 … that’s the only 3-digit number that’s both a perfect square and a perfect cube.”

“Wait … how you so sure of that?” Yona asks.

“Because any number that’s both a perfect square and a perfect cube must also be a perfect sixth power. And since 1 to the sixth is 1 and 2 to the sixth is 64, both of those are too small. But 3 to the sixth is 729, and anything bigger is just too big.”

Gallus turns to Yona. “No offense, but Ocellus might have both of us beat.” The yak nods in solemn agreement.

“OK,” Ocellus continues, “so Yona couldn’t have thought the first two answers were both ‘yes’. On the other hoof, she couldn’t have thought the first two answers were both ‘no’ … there would be far too many possibilities, no matter how the other questions were answered. For example: any of 101, 111, 131, 141, and so on all have identical first and last digits, and all are less than 200. Also, any of 202, 212, 222, 232, and so on all have identical first and last digits, and all of those are greater than 200. And if the first and last digits are not the same, that’s just so many more to pick from. And all that’s to say that Yona must have thought the number was either a perfect square or a perfect cube, but not both.”

“Yona getting confused … but Ocellus right; Yona did see different colored gems for first two answers.”

“That’s still a lot of possibilities, though,” Gallus counters. “I mean, there’s 21 perfect squares with exactly 3 digits.”

“True,” Ocellus admits, “but only three of those have identical first and last digits.”

“But how would you know that a ‘yes’ answer to that question went with the perfect square?”

“Because if it went with the perfect cube, then either Yona would have ‘known’ the number was 343 … that’s the only perfect cube that satisfies that condition … or she never would have been able to figure the number out with only that last question. So instead, there’s only two possibilities at this point: either Yona thought your answers were ‘yes’-‘no’-‘yes’, or she thought your answers were ‘no’-‘yes’-‘no’. In either case, there’s only 3 possibilities … with the first case, the number could only be 121, 484, or 676; with the second case, the number could only be 125, 216, or 512.”

“Incredible,” Yona states in disbelief, “Yona did pick one of those numbers. But … how you know which one?”

“Well, in either of those cases only one possibility is less than 200 … so that must have been what you thought.”

“But that’s still not enough for you, right?” Gallus asks. “I mean, for all you knew, Yona could have answered either 121 or 125.”

“Indeed,” Ocellus replies, “but whichever possibility Yona thought, the other one must have an option greater than 600 … because you said as much.”

“… I did, didn’t I?”

“Yep … so now I know everything: Yona must have read your answers as ‘no’-‘yes’-‘no’-‘no’ and thought the number was a perfect cube less than 200 … specifically, she must have thought the number was 125.”

“That WAS Yona’s guess!” she exclaims in amazement.

“But instead,” the changeling continues, “Gallus actually responded with ‘yes’-‘no’-‘yes’-‘yes’, indicating a perfect square greater than 200 with identical first and last digits … either 484 or 676. And since you said the number was greater than 600, it must have been …”

“… six hundred seventy-six,” Gallus slowly concludes, still stunned at Ocellus’s reasoning.

The sound of a door opening causes all three to turn. Smolder enters the foyer and makes her way towards the table.

“Sorry Gallus,” she says as she picks up her bag of gems. “Just had a craving for a late-night snack.” She notices three emeralds and a ruby on the table. “So what’ve you all been up to?”

“Trust me,” Gallus replies, “wait until morning before asking that.”

Author's Note:

Originally, I posted that Gallus's number was greater than 500. That would have led to an alternate solution: Yona could have been thinking the number was 121 (perfect square, NOT perfect cube, same end digits, less than 200), with Gallus actually thinking 512 (NOT perfect square, perfect cube, different end digits, greater than 200).

Comments

[AstralMouse]

Yay, I got it. That was a fun one.

[Eroraf86]

Huzzah, three in a row! Sadly, I must take a break now. I shall finish the rest later this evening.