//------------------------------// // Puzzle 15 solution // Story: School of Logic: a Collection of Puzzles involving the Student Six // by Coyotek4 //------------------------------// D R D O D – x o D R S R R – o o o O S S E O – x x R O O S T – x o o S E T R E – o o T S O T D – x o [one clue wrong] The students note the six clues scrawled into the upper-left portion of the chalkboard; the remainder of the board filled with an alphabet soup’s worth of letters, squiggles, arrows, and other assorted ‘short-claw’ notation. Atop the desk are papers upon papers of additional scribblings. Their attention shifts from the desk and chalkboard to Smolder herself, still wide-eyed and twitchy as she explains her motives to the others. “Dragons competitive, must be best, so I must be best at solving, and pretty fashion, I like pretty fashion, so I put two and two together to get four, or maybe two more for six, so six clues, but I figured out which one wasn’t really a clue, and I figured out the whole thing. Yes, oh yes I did!” Gallus reaches out a talon to console Smolder. “It’s okay … we came here to see all you’ve done, and we’re very, uh, impressed, with the work you put in tonight. Maybe come back to the dorm and get some rest, and tomorrow you’ll be—” “DON’T TOUCH!” Smolder snaps, as Gallus quickly recoils. “Sorry, sorry, out of line there, but really, it’s all so simple now, I just have to explain it, while it’s all still fresh in my mind and spread out here. Allow me this, and everything will be revealed.” The others turn to each other, then back to Smolder. “Go ahead,” Ocellus calmly beckons. “Good, good. So anyway I said to myself, I said ‘You have to start somewhere’ and so I did, I started by making a simple assumption that some clues were right and see where that led, and what better ones than the ones that give the most information, so I assumed that clues 3 and 4 were correct, because one places two gems exactly and the other places three gems with one correct, so I started there, you see?” “Yona see,” the yak replies apprehensively. “Great, I knew I could count on you seeing this, so anyway if clue 3 is correct then exactly two of those five gems are correct and in the right place but which ones … yes, which ones, indeed … so I have to just go through all of those possibilities and see where each led. So it could be that both onyxes are right, or both sapphires, or one of each, or maybe the emerald with exactly one of the others, so I went through all of those while assuming clue 4 was also correct. Got it?” “Uh, yeeeah, we got it,” Silverstream carefully replies. “Good, so start with both onyxes which would then be in spots 1 and 5, while there also wouldn’t be any sapphires or emeralds, but then by clue 2 or clue 5 there would have to be a ruby, since they’re not both false, but if there’s a ruby and two onyxes then by clue 4 there’s no topaz, and that means that clue 5 is invalid. But that forces clue 2 to be valid, and that means there has to be a diamond since there’s no sapphire and three rubies can’t all be in incorrect places, but that means there’s a ruby, an onyx, and a diamond in the solution and that contradicts clue 1 so there can’t be two onyxes. See, we’re on our way!” “Yes we are,” Gallus states. Right to a padded cell … “So it can’t be two onyxes, but what about two sapphires that would have to be in spots 2 and 3, while there then wouldn’t be any onyxes or emeralds? Glad you asked: by clue 4, there would then have to be a ruby and a topaz, which means clue 5 is invalid, but then clue 1 is valid and there’s a diamond somewhere and that contradicts clue 2 so there can’t be two sapphires, either. Now what about having exactly one onyx and one sapphire. Here’s where it gets complicated!” The group blinks in unison. “This is where it gets complicated?” Sandbar asks incredulously. “Oh my, yes, because now all I know is that there’s no emerald, as well as no second onyx or second emerald. But since either clue 1 or clue 2 has to be valid, there has to be either a diamond or a ruby in the solution. So we break it down into cases: Case 1 and Case 2. Now for Case 1, assume there’s a diamond in the solution, so no ruby. Well then of course clue 2 is invalid, but then so is clue 6 since now there can’t be a topaz but clue 4 implies now that there is a topaz, so we can clearly disregard Case 1.” “Are you sure you don’t want to take a break?” Ocellus soothingly asks. Smolders pays no mind as she continues. “But what about Case 2? Well now there’s no diamond and still no topaz, to go with no emerald, no second onyx, and no second sapphire. All that’s left is to have three rubies to go with the one onyx and the one sapphire, so clue 2 has to be invalid. All I had to do now was check all the combinations that existed.” “You mean permutations, right?” Gallus corrects. “Yes of course I did but thank you anyway so now there’s RRROS and RRRSO and a bunch of other ones but that would take forever right? Of course it would but then I remember clues 3 and 4 which combined force a ruby in spot 1 and then by clue 1 I know the onyx can’t be in spot 4 so another ruby must be in spot 2 and that means the sapphire is in spot 3 and the onyx is in spot 5 leaving the last ruby for spot 4, so RRSRO.” Silence pervades for several seconds. Gallus opens his mouth to speak, when— “I KNOW, RIGHT? That CAN’T be right since now clue 6 is false too, so all that proves that there can’t be exactly one onyx and one sapphire! So now there has to be an emerald, and that makes things a lot easier.” Everyone nods and murmurs agreement. Satisfied, Smolder continues on. “So now that there’s an emerald, can there also be an onyx? If so, there’s no sapphire and if clue 2 were valid there would have to be two rubies and a diamond but that goes against clue 4 so clue 2 has to be the invalid one now, but meanwhile by clue 4 either the ruby is in spot 1 or the topaz is in spot 5 and in either case an onyx has to be in the other spot while an emerald has to be in spot 4. Now the first of these would force a second ruby into spot 2 because of clue 1 but then by clue 6 the only spot left that could be correct is spot 3 which would be a second onyx which we know can’t be, so that rules out the ‘ruby in spot 1’ case, but what about the ‘topaz in spot 5’ case? Well now clue 6 forces either a sapphire into spot 2 or a second onyx into spot 3, but we know there’s no sapphire or second onyx. And just like that, there can’t be an onyx and an emerald.” “What about a sapphire and an emerald? With no onyx, clue 4 forces there to be a ruby and a topaz, so clue 5 has to be invalid, and if clue 1 is valid, there has to be a diamond too, and that goes against clue 6, too, so there can’t be a sapphire and an emerald. So you see what all this means, right?” … “IT MEANS CLUES 3 AND 4 CAN’T BOTH BE VALID!!!” … “How long did it take you to get this far?” Sandbar asks. “Oh about three hours, but now everything gets a lot simpler, because now I know that clues 1, 2, 5, and 6 are all valid, and in particular clue 2 is valid so there’s at least one or two rubies in the solution. Could there be only one ruby? Then there would also be a diamond and a sapphire and by the other ‘valid’ clues there wouldn’t be any emerald, onyx, or topaz, so clue 4 would be invalid and clue 3 would have to be valid with both sapphires correct, but that is counter to clue 2, so there has to be two rubies in the solution, but not three since they couldn’t all be incorrectly placed.” “Yona worried for Smolder sanity.” “Why? I’m perfectly fine, great even, because now I know that there’s two rubies AND I know they’re in spots 1 and 3. And since clue 1 has one gem correctly placed, it’s either the onyx in spot 4 or the diamond in spot 5. And now I can prove that clue 3 is the false one: if not, then there would have to be a sapphire in spot 2 and either an emerald in spot 4 or an onyx in spot 5, so either RSROO or RSRED. The first case can be ruled out by clue 3 itself: it has a sapphire and two onyxes, which is one gem too many; the second case can be ruled out because it goes against clue 5. And that means clue 3 is the false clue!” “It’s all downhill from here. From clues 5, there’s can’t be a sapphire and a topaz to go with the ruby, so from clue 4 there has to be an onyx, and now from clue 1 there can’t be a diamond, so by clue 2 there has to be a sapphire. And since the ruby in clue 1 is in the wrong place, the onyx must be in spot 4, and now by clue 6 the sapphire must be in spot 2. All that’s left is spot 5, and it can’t be a diamond, emerald, onyx, ruby, or topaz; it must be a sapphire. And that is the final solution:” “R-S-R-O-S” Smolder takes a number of deep breaths upon completing her reasoning. “And now I can go to sleep.” She then collapses to the ground. “Wha … where am I?” The young dragon’s eyes flicker open. She lifts herself up from her bed and looks around, finding herself in her sleeping quarters, a pale moonlight giving slight illumination to the room. “How did I end up …” She pauses upon seeing her fellow classmates asleep on the floor, atop a number of blankets and cushions. “Oh …” she says to herself, before turning back in. “Dragon feeling normal?” Yona asks the following morning. “Yeah … thanks again, everyone, for putting up with … whatever that was last night.” “No worries,” Sandbar states. “But I’m still curious … why?” “I dunno for sure … I guess with all the puzzle-solving we’ve been doing, and particularly what I’ve seen from the rest of you, I felt like I hadn’t really proven my own abilities. I kept reminding myself of Dragonlord Ember, and how she became ruler through intelligence and savvy rather than brute force. But she’s still competitive, like all dragons … and I wanted to prove to others, and to myself I guess, that I was capable of solving something like this.” “But why this puzzle?” Ocellus asks. “Well … you don’t all know this, but there’s a part of me that really … likes fashion and girly stuff. I wanted to impress professor Rarity with this, really blow her away.” “You blew all of us away,” Gallus states. “Even if I thought I could solve this one, I never would have had the patience to see it through to a conclusion.” “Yeah … thanks, everyone. But had I known, I probably wouldn’t have chosen this puzzle,” she says with a laugh. “Hopefully today will be less strenuous.” “I think professor Rainbow Dash has more buckball planned for today,” Silverstream says. “Lot of running and flying around, getting the heart pumping.” Yona chuckles. “Yep. Good break from strenuous activity!”