You laid down the original panty. It's shaped like an isosceles right triangle. It wouldn't be a general proof, but who cares! Nothing is cooler than partially proving the famous theorem with panties. Serval will definitely approve.\n\n<img src="https://image.prntscr.com/image/eiDrQ9XHQVa0WtblgNmazg.png">\n\nThe top side is the hypotenuse.\n\n[[put more panties down]]
<img src="https://image.prntscr.com/image/Y9OHISuZQl2DwXaBYsdieQ.png">\n\nYou approached the blue critter and heard it sputters: "Fatal error. Right triangle computation method required. ~droning-sound~ Fatal error. Reboot sequence needed. Right triangle computation method required to reboot."\n\n[[smack the thing with serval]]\n[[show the thing a triangle]]
"Wooww!" Serval cheered with joy. "You were so cool, Miss Bag!"\n\nFennec admitted defeat. "You are definitely very smart. Maybe you can also deal with the strange blue thing over there. It was spouting some weird riddles at me."\n\nRaccoon pointed over to some lush grass near a grand oak tree.\n\n[[little blue thing?]]
"So, Miss Bag," the serval said. "How long have you been here for?"\n\n"The last thing I remember was exploration inside the perimeter of a volcano," You replied. "I don't know how I ended up outside."\n\n"Cool," the serval remarked. "I mean, that's gotta be Hot in the center! So, what are you going to do now?"\n\n[[go back to the volcano]]\n[[follow serval]]
You and Serval departed from the pond and continued exploring. On your way, you two found a mildly interesting duo of friends. A fennec fox was teasing a raccoon. The raccoon laid on the grass, exhausted from a heated discussion between the fennec.\n\nServal approached the fennec fox and inquired. "Why is she on the ground?"\n\n[[next|continue3]]
You ain't a nihilist, and you are damn sure that you exist in the physical world as a moving and metabolizing organism. "I refuse to believe that ridiculous argument," You said to Fennec. Fennec's rhetoric may be a paradox, but like all mathematical constructs, they can be tackled with induction and logic. "There must be some premise in your argument that is fallacious."\n\n"Then prove me wrong." Fennec said.\n\n[[so be it]]
"That was so cool!" The serval was impressed. You showed your work to Toki.\n\n"Bravo, friends," Toki gave you a fluttering hug.\n\n[[hug serval]]\n[[smile and mentally deny the fact that you "cheated"]]
"You know," you started to think. "If this is truly a computer system, that is enough of a computational device. It'll make a nice turing machine."\n\n"So can we keep it?" Serval asked.\n\n"After I solve its riddle." You determined.\n\nHow should you prove the Pythagorean Theorem?\n\n[[with MORE panties]]\n[[with two squares]]
"Methodology established," the turing machine spoke. "Reboot successful. Welcome to Japari Park, Friends."\n\n"Say," you asked. "Could you give me the square root of pi?"\n\n"Square root function already installed. 64-bit word Pi constant in memory loaded into register. Returned 1.77245385091 to twelve significant digits. Computed in 0.12 microseconds."\n\n"We did it!!" You and Serval both cheered for joy and embraced each other.\n\n<img src="https://image.prntscr.com/image/70320joPQjWqx3sJbtYXRA.png">\n\n[[epilogue]]
You smiled and shaked Toki's hand. Your solution was good enough to do the job.\n\n[[let's move on|continue2]]
<img src="https://image.prntscr.com/image/OaZ33xY8SVqZaJVClI9VlA.png">\n\nFennec gave an eccentric look at us and said, "I told Raccoon that if she wants to run a certain distance - let's say 100 meters, then she must first reach the 50-meter mark, and to reach that, she must first run 25 meters. But to do that, she must first run 12.5 meters."\n\n[[keep listening]]
<img src="https://image.prntscr.com/image/ZkIS2rBlTJKds3tDWAJk9A.png">\nYou wake up in a savnnah and find yourself dressed with a safari hat and a white bag, ready to explore!\nSuddenly, a petite girl dressed like a serval cat approachs you.\n\n[[Say hi]]\n[[Look her over]]\n[[run away]]\n
"Toki," you said after giving the problem a deep thought. "Can I lay down two initial rocks however I want?"\n\n"Of course," Toki replied. "As long as it roughly covers the wetland."\n\nYou place the two rocks to make two orthogonal rays, which is a fancy way of saying the two rays are 90 degrees apart. On your way back to the pond, you also picked up a hardy stick that was laying on the ground.\n\n<img src="https://image.prntscr.com/image/5w4aWiKpSrC0mLD0j77Bxw.png" width="500">\n\nThen, you unpacked your bag and grabbed a rope. "Serval, could you hold on to this end of the rope and stand on the rock near the pond?"\n\n"No problem," the serval held onto the rope's end and obediently stood near the pond's rock.\n\n"Make sure you don't move, Ok?" You told her. "I'll show you something cool."\n\nYou started from the upper left rock and walked in a quarter-circle, making sure the rope is stretched the entire time and used the stick to mark the ground as you went. You stopped once you reached the lower right rock.\n\n[[so far, so good]]
"I don't have much to do," you said. "And my memory is still fuzzy. Can I just follow you for now?"\n\n"Naturally!" the serval spoke with joy. "I'm actually going to meet a friend. She runs a cafe." The serval grabs your hand and walks toward the perimeter of the volcano anyway.\n\n[[continue walking|volcano1]]
Raccoon seems to be having an existential crisis right now.\n\nServal seems to be worried, though. "That's not good! I really want to be real friends with you, Miss Bag!!" She held onto your hand tightly with both of hers.\n\nYou didn't want to confuse the sweet and innocent Serval. So you decide to finally put this amusing paradox to its rightful place, with beautiful logic and reasoning!\n\n[[bring it on|continue4]]
"Preposterous!" Raccoon exclaimed. "Help me out here, friend with the Bag and friend with the spotty dress." She looked at you and Serval.\n\n"Weeelll..." You decided to play along with Fennec's prank for now. "Fennec does have a point. This is all an intrinsic illusion that you've having right now. This whole world is a lie, and you're just dreaming in some null and motionless void."\n\n[[its joke ( ͡° ͜ʖ ͡°)]]\n
Fennec continues. "Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus Raccoon has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so she can never reach her goal. In general, any Friend who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion."\n\nRaccoon sprang up and gave everyone an incredulous look.\n\n<img src="https://image.prntscr.com/image/q38xD-ncRg2UyFd_O5KYuQ.png">\n\nRaccoon is clearly disappointed that she won't be going anywhere.\n\n[[hahaha, is that so?]]\n[[my bullshit-o-meter is off the charts]]
"What's a multi shell flannel?" She bats an ear. "Is that what you're holding? I'll just call you Bag, then!"\n\nThe serval shakes your hand. "Nice to meet you, Miss Bag. I don't bite friends as cute as you!"\n\n[[Well then..|continue1]]
PREMISE TWO\n- crossing an infinite number of midpoints in a finite time is impossible.\n\nYou realized that premise can be rephrased to:\n\n'Taking an infinite sequence of real numbers (e.g. sequence of midpoints) and summing them up (crossing all their distance) will not yeild a finite result.'\n\nIs this premise [[always true]], [[never true]], or [[sometimes true]]?
A series that diverge means it goes AWAY from a finite value. That's not what you are looking for...\n\n[[back|previous()]]
Both of you traveled to the upper left rock and repeated the process of generating another arc intersection. Then, you placed a rock on each intersection.\n\n<img src="https://image.prntscr.com/image/vHjcgNfIStaAT0zq3t5B4g.png" width="500">\n\nFantastic! The angle trisection for a right angle has been completed.\n\n<img src="https://image.prntscr.com/image/1U8OknJkS__vjOIZO9sONA.png" width="500">\n\n[[give yourself a pat on the back]]
PREMISE ONE\n- there are an infinite amount of midpoints between any two arbitrary points in Euclidean space.\n\nThat premise is true without a doubt. There are a systematically countably infinite number of rational numbers between two points in a number line. Any division of a rational number will yield another unique rational number, and that process can go on forever.\n\nIt seems you must [[tackle the other premise|the second one]].
You tried to scuttle away.\n\n"Don't go!" The serval pounced on you.\n\n"Please don't eat me!" You yelled for your life.\n\n"I'm not going to eat you," the serval giggled. "Say, what are you? I've never seen something like you before."\n\n[[A Bag.]]\n[[A multicellular mammal.]]\n[[A friendly genetically modified organism.]]\n
You decided to use proof by dissection, which relies on the use of two squares, one inscribed inside the other.\n\nThere's nothing paper and chalk can't solve. You started drafting the proof:\n\n<img src="https://image.prntscr.com/image/RI7KuQRtTn6Zpp6qzxC2yA.png">\n\n<img src="https://image.prntscr.com/image/llRxIDRWRXeVivzVXk9kfw.png">\n\n[[next|algebra pyth]]
<img src="https://image.prntscr.com/image/yZBLxnhqQw6Bn7JwWeod0w.png" width="500">\n\nWhat should you tell serval to do now?\n\n[[Play with the rope]]\n[[Walk to the upper left rock]]\n[[Walk to the lower right rock]]\n
<img src="https://image.prntscr.com/image/87HqdIw0TnKEBF8cChdtAQ.png">\n\nAuthor's note: if I have had more time, I would have written a single bonus problem to those who made less than or equal to 1 mistake in the entire story, to complete the entire set of ancient greek geometric problems of antiquity.\nFan art illustrations are not mine.\n\nI hope you enjoyed this interactive fiction story.
"In fact," you continued to present evidence. "Consider the problem you posed." You swiftly drew some markings on your straight-edge.\n\n<img src="https://image.prntscr.com/image/RLJhHXAPToGbah-TexpL5w.png">\n\n"The ratio between successive terms is 1/2. The first term is 0.5, and the infinite series converges to 1." You concluded.\n\n[[torture Fennec more with the theorem showing which geometric series converge to a finite sum]]\n[[let's move on for now]]
If this premise is always true, then there's no point in analyzing it. Better examine the [[other one|the first one]]?
Serval pounced on the thing.\nIt toppled over and struggled.\n\n"I don't think that is the answer to his riddle." You said.\n\n[[try something else|show the thing a triangle]]
"Miss Bag," the serval inquired. "That seems like a simple task. Why didn't they work it out?"\n\n"They probably did," You replied, still getting used to your nickname. "I think whenever one of the ibis finished laying down four rocks to mark the four rays, another ibis thinks it's laid down incorrectly and disagrees. It's a repeating cycle."\n\n"mhmm," Toki chirped. "Friends, if you can show us a method and convince all three ibis that it fairly divides the sectors, we'd deeply appreciate your mediation."\n\n[[challenge accepted]]\n[[uh.. I'm not so sure..]]
"I'm... a friendly GMO.. like all living things," you said proudly. "Eras of evolution have branched off my ancestors into who I am today." Your expertise in biology is paying off. "In fact, our ancestors migrated from this continent. You and I are related in this environment."\n\n"I'm not sure what that means," the serval said. "But a friend of the savannah is my friend! What should I call you?"\n\nYou tried remembering your name, but you're still frizzed from this bizzare outcome. You simply held up your bag. "Call me Bag for now."\n\n[[that worked out strangely well|continue1]]
"While it's true that a sequence such as 1, 2, 3, 4, 5, ... that goes on to infinity will yield an infinite sum when added together," you started to explain. "There are a class of infinite sequence that, when added, will yield a finite result."\n\n"Oh? Enlighten me." The Fennec sits down and puts a palm under her chin.\n\n[[Enlighten her]]
"I'm sorry to say this," you began. "But the new square's length will have to be expressed in some term of pi, which is transcendental."\n\n"Do you need a pie? I can bake it for you," the oblivious alpaca inquired. "Take one for your trip, too, dear."\n\nThe serval licked her lips, thinking of a meat pie. Preferrably one with rodent-fillings.\n\n"Oh, my bad," you said. "What I meant is the length of the new square is unsolvable in Euclidean space. In other words, it cannot be done with only a straight-edge and a compass; unfortunately that's all I have in my bag."\n\nAfter a short pause, you felt bad for giving a 'no-solution' to the alpaca, so you continued. "If I can get my hands on a computing device, I promise to come back and help you."\n\n"Ala! That's nice of you." The alpaca responded sincerely.\n\n[[Quest for the computing device!]]
"Leave it to me," you said with confidence. You reached into your bag and pulled our your trusty compass and straight-edge, as well as a sheet of blue drafting paper.\n\nYou wrote out the equation:\n(side)^2 = (pi) r^2\n\nand solved for the length of the side in terms of the radius.\n\nside = sqrt(pi) * r\n\n"Hmm, it seems I was overconfident," you suddenly realized as you looked at the ghastly coefficient in front of radius.\n\n"What's wrong?" the serval asked.\n\n"There's no way I can draw a line segment that is increased by a factor of the square root of pi from the radius..." you explained midway, then changed your wording. "I simply don't have the proper tools to do it."\n\n"What kind of tool?" inquired the serval.\n\n"Some kind of processing unit for computation," you replied.\n\n"I'm sure it'd be fun to look for this purr-sessing yuunito," the serval butted in. "Come on, let's go!"\n\n[[Quest for the computing device!]]
What strategy should you use to approach this?\n\n[[guess and check]]\n[[Taylor's theorem for power series]]\n[[The geometric series]]\n\n<<if visited("guess and check", "Taylor's theorem for power series", "The geometric series")>>\s\n[[one must cheat to solve this problem]]\n<<endif>>\s\n
Taylor's expansion series allows you to expand a sophisticated function into a sum of many simpler functions.\n\nUnfortunately, you know of no function that describes how to trisect an arbitrary angle... at least not now..\n\n[[think again?|previous()]]
<img src="https://image.prntscr.com/image/52mHuHgoRwaU_3dTTZBpjQ.png">\n\n"Look, turing machine." You spoke to the blue thing. "The hypotenuse (c) is the waist line. It makes a square torso that has an area of 4 panties. c^2"\n\nThen you pointed to the non-hypotenuse legs of the triangle. (legs, get it?) "The left leg makes the left thigh that has an area of 2 panties. The right leg makes the right thigh that has an area of 2 panties. Add them together, and you get the area of the hypotenuse square."\n\n[[cool, huh?|continue6]]
"Where am I?" You exclaimed.\n\nThe serval looked at you with excitement. "You can talk!" she replied. "You are in Japari Park!" She answered with joy. "I haven't seen any animal like you before. What are you?"\n\n[[A Bag.]]\n[[A multicellular mammal.]]\n[[A friendly genetically modified organism.]]
"I'm.. a bag," You answered, and you have no idea why you said that... perhaps because you're holding a big white bag. Plus, you had trouble remembering your real name. "Call me Miss Bag."\n\n[[OK.|continue1]]
An arithmetic sequence of numbers in which each differs from the preceding by a constant quantity\n(e.g., 3, 6, 9, 12, etc.;\nor 5, 3, 1, -1, -3 etc.).\n\nThe sum of an infinite arithmetic sequence will either diverge to infinity or negative infinity. It won't converge at all.\n\n[[try again.|previous()]]
The rope is not long enough for her to travel over there.\n\n[[back|previous()]]
You arrive at the perimeter and found a cafe. Serval went inside, and you followed her and met an amiable Suri Alpaca. She seems shy, but welcomed you two.\n\n<img src="https://image.prntscr.com/image/Zy6waWdYS2KQbCbKFdVdyw.png">\n\n"Hi, ladies," the alpaca greeted. "Care for some tea?" She lifts up a tray of an assortment of brewed tea. "In exchange, would you mind helping me with a dilemma of mine?"\n\n"Sure thing!" The serval doesn't seem to care much for tea, for she simply gave hers to you.\n\n"The circular kettle lid I've used has broken," the alpaca explained. "I've made a new kettle, but I ran out of metal for its new square lid. I could use the volcanic furnace to reforge the old lid, so it'd be nice if you can draft a blueprint of the square mold."\n\n"So we simply need to make a square mold that has the same area as the circle lid," recapped the serval. "That should be easy, right, Bag?"\n\n[[a piece of cake!]]\n[[actually..]]
As adorable as that would be, you need to preserve the length of the rope until the angle trisection is complete.\n\n[[Nyyaaaa|previous()]]
You paused to think for a moment and considered the two premises of Fennec's argument:\n\n- there are an infinite amount of midpoints between any two arbitrary points in Euclidean space.\n- crossing an infinite number of midpoints in a finite time is impossible.\n\nIf you can show that one (or more) of these premises are false, then the entire argument is unsound. Even if the conclusion follows logically from the premises, it will not hold with faulty premises.\n\nWhich premise should you tackle?\n[[the first one]]\n[[the second one]]
You carefully took out a spare panties.\n\n"Optic sensor detected right-triangular object," the blue thing uttered. "Hypotenuse is square root of sum of side squared side squared ~droning-sound~ Need algorithm. Need proof."\n\n[[comply?|continue5]]
"I believe that is never true!" You pointed at Fennec and shouted.\n\n"Are you sure about that?" Fennec replied mischievously. "Let me give a counter-example. Consider the sequence 1, 2, 3, 4, 5, ... and it goes on infinitely. The more I add them together, the bigger and faster it grows. Clearly, if I add them infinitely, I would get an infinite sum."\n\nYou recoiled in defeat. Fennec's objection to your objection was flawless. You'll need to come up with a better one then that.\n\n[[try again.|the second one]]
<img src="https://image.prntscr.com/image/QfZVjnK6RHigaUhmx1aPBg.png">\n\n"Oh, friends.." the crested ibis muttered. "You can call me Toki."\n\n"Hi, Toki!" the serval was fast to give her salutations. "Why are you feeling down?"\n\n"Three of us ibis are disputing over which sector belongs to whose territory," Toki said. "since we always come back to this pond for water, they figured that we can start with this rock near the pond and then use four more rocks to extend four imaginary lines outward that divides the wetland area into three equally large angular sectors."\n\n<b><u>So it's basically cutting a portion of the pie into equal pieces.</u></b>\nYou can already see what kind of conflict Toki ran into.\n\n[[next]]
"Hey, it mentioned tryangels," Serval said. "Ya got any, Bag?"\n\nYou looked into your bag and blushed. "Well, I do have triangular objects here."\n\n[[what kind of triangular object? ( ͡° ͜ʖ ͡°)]]
"This class of series is called...\n\n[[Infinite Arithmetic Series]]\n[[Converging Arithmetic Series]]\n[[Converging Geometric Series]]\n[[Diverging Geometric Series]]
The geometric series is a sum of a sequence with a constant ratio between successive terms.\n\nYou don't know why you thought of that. Perhaps it will come in handy later, but for now, you see no way to apply this mathematical concept to angle trisection.\n\n[[think again?|previous()]]
"So," you said to Fennec. "What you're saying is that taking an infinite sequence of numbers and summing them up will not yeild a finite result."\n\n"Absolutely." Fennec chortled.\n\n[[OBJECTION!!]]
It'll take ages to reach a configuration that'll satisfy all three ibis by simply guessing and checking repeatedly. Plus, you'll just bore everyone out.\n\nYou need a systematic and scientific method that appeals to logic and reasoning. Otherwise, you will be unable to quell each of the ibis' individual biases that will inherently result from ad-hoc endeavors.\n\n[[think again?|previous()]]
An arithmetic sequence of numbers in which each differs from the preceding by a constant quantity\n(e.g., 3, 6, 9, 12, etc.;\nor 5, 3, 1, -1, -3 etc.).\n\nThe sum of an infinite arithmetic sequence will either diverge to infinity or negative infinity. That's not what you're looking for.\n\n[[back|previous()]]
She has a pair of tall and fluffy ears on top of her blonde hair, and a spotted dress with a cute ribbon. Adorable, even.\n\nThe front of her hair has some black mark that makes a letter 'M'. You wonder what that stands for, since "serval" doesn't start with an M.\n\nThe serval looks back at your intently, to your chagrin.\n\n[[Say hi]]\n[[run away]]
You gave serval a hug.\n\nShe purred and hugged you back.\n\n[[That was sweet.|continue2]]
"I'm going to check out the volcano and see what clues I can find there," you planned.\n\n"Rad!" the serval said. "Can I come too?"\n\n"A'ight. Let's go"\n\n[[to the volcano|volcano1]]
You're not sure where serval is taking you. You're not even sure if a powerful computational device even exists concretely in this vast savannah. Buuut you follow her anyway because you have nothing better to do; plus, at least you and the serval are putting in some effort, right?\n\nAlong the way, you two arrive at the edge of a pond.\nNear some rocks, you see a beautiful figure of a crested ibis. She seems to have noticed the serval's bright stripes and spots too.\n\n[[greet the ibis]]
You told Serval to come to where you are standing right now, then told her to stay there. Then, you carefully held onto your end of the rope, making sure the length was exactly same as before, and walked until the rope is fully stretched. You move radially until the stick has marked an arc intersection.\n\n<img src="https://image.prntscr.com/image/AtIKubiDT_qZxwW8RkvYYA.png" width="500">\n\n"Great!" you rested a little. "One more step to go."\n\n[[do the other side]]
You may be right to be skeptical of any euclidean solution to this problem, but\n\ncome on, man, you gotta at least try before you give up.\n[[Ok. Fine.|challenge accepted]]
Serval tugged on you. "Is it true, Miss Bag? Are we all just a figment of imagination?" She looks at you with innocent eyes.\n\nFor both Serval's and Raccoon's sake, you decide to put Fennec's prank back to rest. She's had enough fun already, and now it's time for you to shine!\n\n[[get your neurons fired up|continue4]]
Then you simply applied FOIL and algebra.\n\n<img src="https://image.prntscr.com/image/9jRy0DdvRviz9W0dKsJmIA.png">\n\n[[Q.E.D.|continue6]]
two girls go on an adventure.\nparody by Ching Lam Yung
...converging geoemtric series." You said with great confidence.\n\n"Do elaborate." Fennec said.\n\n"Consider the following." You took out a sheet of paper and drew some squares in it, and used some chalk to color some parts.\n\n<img src="https://image.prntscr.com/image/-VLKNMw0QAOkwKmZeLieXw.png">\n\n"Here you see the main outer square that contains smaller squares," you started to elaborate. "The outer square has side length 1. We start with the bottom left purple square. The length of its side (1/2) is midway of the outer square's. The next smaller purple square in this sequence has side (1/4), which is another midpoint."\n\nYou took a deep breath and continued. "Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). As you can clearly see in the diagram, <u><i>I didn't need an infinite amount of paper to contain all the purple color</i></u>, because the area sums to a finite amount! One third of the area of the original main square, to be exact."\n\n[[checkmate, Fennec.]]
A moment of history flashed into your mind and you remembered that the ancient greek problem of general angle trisection was algebraically proved impossible by Pierre Wantzel in 1837.\n\nEven though you don't know of a universal method for trisecting an angle, you do know of two special cases that can be easily trisected. You decided to use this to your advantage in solving the crested ibis' dilemma.\n\n[[Let's get your hands dirty]]
You wrote the theorem and taped it onto Fennec's head.\n\n<img src="https://image.prntscr.com/image/cvg8PIrZRDGWO1j3oyvTpA.png">\n\n[[move on|let's move on for now]]