//------------------------------// // Puzzle 14: Balancing Act // Story: School of Logic: a Collection of Puzzles involving the Student Six // by Coyotek4 //------------------------------// “ ‘Logic bomb’? So, these ponies would black out over a sentence?” Coccyx nods as Smolder processes the answer to her question. “Nothing personal, but I couldn’t see any dragon passing out over a bunch of words.” “A by-product of their way of reasoning,” Coccyx assures. “From the beginning, Utopia was created solely for unicorns and Paradise for pegasi. Even after unicorns began settling in Paradise, their pride and sense of loyalty persuaded them to maintain an allegiance to the town of their ancestors’ origin. Hence, no unicorn would ever admit to coming from Paradise, and vice versa. From there, notions of truth and falsity spread organically, up to the point when Twilight first made contact.” Yona asks for clarification: “So, if unicorn claim to be from Paradise, that caused fainting?” “Before Elysium came into the picture, yes,” Coccyx replies. “At that time, all unicorns claimed to come from Utopia and all pegasi claimed to come from Paradise. For that matter, once one factors in the faux-alicorns, all Utopian claimed to be unicorns and all Paradisians claimed to be pegasi.” “But Elysium changed all that, right?” Sandbar asks. “In many ways … now a unicorn could claim to be from Paradise, if in fact she was from Elysium. But no unicorn would claim to be from either Paradise or Elysium.” Coccyx’s last statement draws quizzical looks from the class. “You just said a unicorn could claim to be from Paradise if she was from Elysium,” Gallus says. “And I assume a unicorn could also claim to be from Elysium, if she was actually from Paradise.” “Yes, and yes,” Coccyx replies. “But it’s not possible for a unicorn to claim to be from either Paradise or Elysium? That makes no sense!” “I realize it sounds weird, but it’s true. Suppose a unicorn did say ‘I come from either Paradise or Elysium.’ Assuming that unicorn came from one of the three towns, which one could it be?” Gallus mulls over the question. “… if the unicorn was from Utopia, then she would be truthful, but her statement would be a lie. So the unicorn would actually have to be from Paradise or Elysium, making her statement true … but unicorns from those towns always lie … huh … that’s weird.” “And believe me, I found out the hard way,” Coccyx continues. “While posing as a pegasus, I commented to a townspony that I wasn’t from Paradise; that pony dropped like a rock.” He smiles as he continues. “But they don’t have that problem anymore.” “Why not?” Silverstream asks. “Because of changelings like myself,” he proudly proclaims. “Now that the townsponies are aware of our presence, they can rationalize that such a comment could only come from a changeling. Rather than fear our presence, the knowledge of our being there prevents similar occurrences from happening.” “Wow,” Ocellus admits, “it’s a logic-based symbiotic relationship.” “In a sense, yes. We fill the gaps that would normally come from certain statements. Really, we complete the towns.” “Like imaginary numbers!” Yona asserts. Sandbar, Smolder, and Silverstream turn to Yona with curious looks. “It math thing,” she continues. Ocellus re-addresses her uncle: “So, the changelings are still in good standing with the Alicorn Society?” “Indeed … in fact, new tests have been created that incorporate our presence. Recently, one particularly difficult challenge was created for new recruits. I would like to see how you as a class respond to it.” “Harder than the last one?” Silverstream asks. “Well … you be the judge,” Coccyx says before explaining the challenge: “You are introduced to four alicorns. You are informed that one of them is a changeling with no particular hometown. The other three alicorns consist of one unicorn, one pegasus, and one earth pony; in addition, each one comes from a different one of the triple towns. Each alicorn will make one or two statements, with the usual rules of truth and falsity applying: only a Utopian unicorn, Paradisian pegasus, or Elysian earth pony will make truthful statements, while ponies of the other combinations will lie.” “As for the changeling, he or she will either be truthful or lie … whichever is less prevalent among the others. So if two or three of the others are truthful, the changeling will lie; if two or three of the others lie, the changeling will be truthful. Think of the changeling as attempting to balance out the veracity of the group.” “I will refer to the four alicorns as A, B, C, and D, as before. They make the following statements:” A: C is a pegasus. B: A is from Utopia. C: D is a changeling. A is a unicorn. D: Only one of us is a liar. B is from Paradise. Ocellus ponders the statements. “That doesn’t seem like enough to go on.” “It’s not,” Coccyx replies. “You are also given two additional facts:” A recruit who took the test before you was then told which alicorn was from Elysium, but that recruit still did not have enough information to solve the puzzle. Another recruit who took the test before you, who knew nothing of the first recruit, was then told which alicorn was the earth pony; that recruit also has insufficient information to solve the puzzle. “And now YOU have enough information to solve the puzzle! Which alicorn is the changeling, and what are the pony types and hometowns of each of the others?”