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

by Coyotek4

Puzzle 20 solution

“OK,” Ocellus begins, “for these five rounds, you’ll each see two Smolders, one of which will be me. Each of us will have a card from 1 to 4, and of course, we won’t have the same card. You’ll have to figure out which of us is really Smolder, as well as which card each of us has.”

The foursome look at each other quizzically, before Silverstream addresses the changeling: “Uh … how is this any different from what we just did?”

“One small change, but it makes a big difference: this time, if I have the 1 or 3 then my statements will be false, and if I have the 2 or 4 then my statements will be true.”

Ocellus’s comment is met be more strange looks. “That exactly how first five rounds went,” Yona points out.

“Oh … yeah, that part’s the same.” Ocellus begins to blush as she continues: “What’s different is—”

“What’s different,” Smolder interrupts, “is that if I have the 1 or 3 then my statements will be true, and if I have the 2 or 4 then my statements will be false. It’s the opposite of Ocellus.”

“That … really doesn’t sound all that much more challenging,” Gallus states.

“Hey, I’m game either way,” Sandbar replies. “Go ahead and start.”

The foursome collectively close their eyes, and the rounds begin:

[In each round, one Smolder appears on the group’s left (L) and one appears on their right (R).]

[Round 1]
L: “I am Smolder. I have the 1.”
R: “I am Smolder. I have the 2.”

It takes only a few seconds for Silverstream to respond: “Well … one is lying and one is telling the truth. I’m sure of that.”

“Shouldn’t be hard to figure out which is which,” Gallus adds. “Which one said something that couldn’t be said?”

Sandbar ponders for a moment … “The Smolder on the right said she had the 2 … but the real Smolder wouldn’t truthfully say she had the 2, so that has to be Ocellus. And, she has to have the 1 or 3.”

“But then Smolder on left truthful,” Yona adds, “so Smolder on left have 1. That only leave 3 for Ocellus on right.”

In a flash, the Smolder on the right changes back into Ocellus. “Not bad. Let’s see how you do on the next one.”

Eyes close, next round begins …

[Round 2]
L: “I am Smolder. Ocellus has the 3.”
R: “I am Ocellus. Smolder does not have the 4.”

Silverstream again begins the thought process: “They each to be somecreature different, so they’re either both truthful or both lying this time.”

“But if both truthful, Ocellus have 3,” Yona points out. “That not right.”

“So both are lying,” Gallus concludes. “Ocellus is on the left and Smolder is on the right. Now as to their numbers …”

“They both lied,” Sandbar states, “so Smolder does have the 4. And since Ocellus can’t have the 3, she has the 1.”

Another flash, and the Smolder on the left reverts to her natural form. “It feels weird to say ‘Ocellus has’,” she admits.

“Meh,” Smolder says. “I know dragons who like to refer to themselves in the third-creature. No big deal.”

“Just big egos,” Gallus says to himself, before quickly backtracking upon noticing Smolder’s death-glare. “Sorry, sorry … old habits … my apologies!”

“Wellllll, you may not be too far off with certain dragons.”

“We’re all cool, right?” Sandbar asks.

“It’s fine,” Smolder says. “Let’s just get back to the rounds …”

[Round 3]
L: “I have the 1. The other card is not the 2.”
R: “I have the 3.”

“Wait,” Yona says, “neither one claimed to be Smolder or Ocellus. This round actually have solution?”

“There has to be a solution,” Gallus states, “… and it begins with the fact that Ocellus is lying.”

“How do you know that?” Silverstream asks.

“Ocellus would never truthfully state she has an odd number, either 1 or 3.”

“Good point,” Sandbar says. “Does that mean Smolder is telling the truth?”

Yona ponders for a moment … “No! Smolder could still be liar; all we know is Ocellus definitely liar.”

“All right,” Silverstream continues, “so which one is Smolder? Could she be on the left?”

Gallus thinks the possibility over. “Suppose she was on the left and telling the truth. Then she does have the 1 … and if Ocellus is lying, she would have to have the 3.”

“But that would make her statement true,” Sandbar adds. “That doesn’t work.”

“So now say she’s lying,” Gallus continues. “Then the ‘other’ card is a 2, which would mean Ocellus is telling the truth … that doesn’t work, either.”

“So that mean Smolder not creature on left,” Yona concludes. “Ocellus on left, and both statements false; that mean other cards is 2.”

Silverstream excitedly wraps up the solution: “That means Smolder is lying and has the 2, and Ocellus being on the left means she has the 3!”

The Smolder on the left changes back into Ocellus as the one on the right comments: “Wow, you guys really are getting good at this.”

“Two more rounds,” Ocellus points out as they begin anew …

[Round 4]
L: “The sum of our numbers is 5. My number is the smaller of the two.”
R: “One of our numbers is double the other. My friend’s card is not the 1.”

Silence pervades for a full minute. Finally …

“The one on the left is lying,” Gallus states.

“How do you figure that?” Sandbar asks.

“If she was truthful, their cards would have to be one even and one odd, and that would force both of them to be telling the truth. But that’s impossible, since the only ways to have one card be double the other is with the 1 and 2, or the 2 and 4 … and neither of those add up to 5.”

“Whoa,” Silverstream says, “that is some high-level thinking there!”

“So where we go from here?” Yona asks.

“What if both of them were lying?” Sandbar asks. “Then they’d still have one even and one odd card, right?”
Gallus nods.

“But,” Sandbar continues, “we already know their cards don’t add up to 5, so the cards would have to be either 1 and 2, or 3 and 4, right?”

“Sounds good to me,” Gallus replies.

“Wait,” Silverstream says, “1 and 2 are out, right? ‘Cause, that would make the ‘double’ statement true.”

“That leave 3 and 4,” Yona adds, “but that make other statement true, about friend card not being 1.”

“So all that means the statements from the Smolder on the right can’t be false; they must be true,” Gallus concludes.

“That means the numbers are either both even or both odd,” Sandbar states. “They can’t be both odd, since one is double the other, so they must be 2 and 4.”

“And since the Smolder on the left lied,” Silverstream adds, “her number is the bigger of the two. And that means—”

“Smolder on left with 4 and Ocellus on right with 2,” Yona concludes.

Ocellus (on the right) reveals herself. “That was amazing! I really thought I had you stumped on that one.”

“That just leaves one final round,” Smolder says. “Let’s see if you can go five-for-five …”

[Round 5]
L: “Either I am Smolder or I have the 3.”
R: “I am Smolder and I do not have the 4. Neither of us has the 2.”

“I don’t like conjunctions,” Silverstream admits. “They make my head hurt with these puzzles.

“We gotta start somewhere,” Gallus states. “What if the one on the left is the real Smolder?”

“That mean one on left truthful,” Yona replies. “Also, one on right lying if that one actually Ocellus.”

“And if she’s lying,” Silverstream adds, “then one of them has to have the 2.”

“But that’s not possible,” Sandbar insists. “Either Smolder would have the 2, and that would force her to lie, or Ocellus would have the 2, and that would force her to be truthful. Neither of those works with these assumptions.”

“So much for figure out which one is Ocellus,” Gallus says. “The one on the left is Ocellus.”

Rather than wait for the final solution, Ocellus (on the left) changes back. “Wow … I really thought this was the hardest one.”

“They still need to figure out the cards,” Smolder points out.

“But if Ocellus truthful, she would have 2 or 4,” Yona says. “That not work here, so Ocellus lying and have 1 or 3 … but not 3, since that make statement true. So Ocellus have 1.”

“Meanwhile,” Silverstream adds, “the only way Smolder could by lying is if she has the 2 or 4. If she had the 2, her first statement would be true, and if she had the 4, her second statement would be true. So Smolder must be telling the truth, and her card has to be the 1 or 3 … but not the 1, since Ocellus has it, so she has the 3.”

Ocellus and Smolder each reveal their cards, confirming the final piece of reasoning.

“That was actually a lot of fun!” Silverstream states. “So what’s next?”

Ocellus shrugs. “There’s not much more I can do with this format? I mean, I could make it so one of us is truthful with 1 and 4 and the other is truthful with 1 and 3, or some other combination, but really, it’s all pretty much the same.”

“Maybe we need two pairs, like two Sandbars and two Galluses,” Yona says. The others chuckle at the thought.

“Maybe we should just have two of all of us,” Gallus says. “That would mess with someone’s mind.”

“But if we did that, who’d be left to actually solve the puzzle?” Sandbar asks.

Ocellus takes all the comments in as her mind races …

Comments

[ANW]

“But if we did that, who’d be left to actually solve the puzzle?” Sandbar asks.

Well technically, Spike is young.

[Jubilee Bloom]

9657989
Interesting...

[son_of_heaven176]

10564623
I wasn't asking you to change anything this time; I was staying that there was another way to get to that conclusion.

But in general, thanks for being quick to respond.