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Wraith Arc: Chapter 9

2023-05-06T06:17:10Z

Agnesinφ: Alas, the oasis of sweets was but a mirage. ( changed 'dessert' --> 'desert')

{{spoilers}}

The chapter opens with a landscape scene of the city. Wraiths swarm the streets as Homura sits on the edge of a building. Her soul gem glows and her time shield appears in her hands. She tries to activate it, but the shield only makes a noise and nothing more. Homura considers the shield, wondering why it was left behind since she has no need to rewind time anymore. In the previous world, she used her time manipulation magic to attempt to save Madoka countless number of times. Once the new world came into being, her shield and the power to rewind time became a thing of the past. Homura is reluctant to part with it, but sees no reason to keep it around and drops the shield into the streets below. As the shield hits the ground, a blast of magic bursts forth from the center of the shield temporarily blinding Homura. A blackhole appears where the shield was. Homura enters and finds herself wandering a vast world, black ‘pins’ dotting the otherwise empty landscape. Homura is uneasy since this place feels like somewhere she’s seen before. She hears a voice calling her name and she begins running and calling out. She hears the voice calling her name once more and Homura is almost certain it is Madoka calling. The floor beneath her gives way into strands of hair and she turns, ready to fire her bow at any enemy that may come at her. Instead she finds herself staring into the right eye of Kriemhild Gretchen.

Back in the city, Homura sits on the ground gasping for breath as the shield quietly lays on the ground beside her. She can’t comprehend what the thing inside her shield is but the feeling it gives her leaves her little doubt as to its nature: that thing is a witch. Now that the shield has accomplished its purpose as a time manipulation device, another dimension was constructed inside it. And in it, the memory of that ‘thing’ still haunts it. There should be no more witches in this new world because Madoka rewrote the universe so she would be the one to fight them for all eternity. Homura raises the shield above her head, ready to smash it, but stops. She realizes if what she saw was really a witch, then the Madoka that must fight witches for eternity must be inside as well. Homura decides it would be better to forget about this completely and proceeds to rewrite her own memories with her magic, burying the memory of the shield and the creature within to the deepest part of her memories. She strongly feels her duty to protect the world Madoka protected no matter how much the world or her magic may change because she is the shield that protects Madoka.

In the present, Homura awakens, a piece of building debris pinning her to the ground. She looks up at the transformed wraith towering above her. Homura has finally remembered everything and realizes that it was the power of the shield that they absorbed that caused the wraiths to mutate and go rogue. However, she is bleeding profusely and doesn’t have her magic back yet in order to heal herself. Close by, Wraith-Madoka lays heavily injured but her hands still hold the ribbon she took from Homura. As she reaches for the ribbon, Homura berates herself for altering her memories so much she doubted the existence of Madoka. Rather than protect her world she ended up causing this horrible disaster instead. She reaches out once more and begs Madoka “please lend me -”, but before she can finish her thought, Madoka takes her hand and holds it. Madoka smiles warmly at Homura and tells her it’s alright now. She told Homura before that she is someone that exists for her. Madoka holds her hands out and Homura is enveloped in light. Homura gasps as she stares at an unknown figure, her looks changed to that of her previous self with braids and glasses. Flower petals swarm around them both. Homura walks closer to the figure and sees that it is herself but different: her twin wears a long white dress, loose hair, and a hollow core with an orb of light in the center. Homura asks her twin what this place is. A hite bird lands on the white-dressed Homura’s hand as she explains that Homura’s consciousness has been temporarily separated and is now in this place, which is the inside of Homura’s soul gem. She reassures her that she is not her enemy and is actually a piece of the power that was stolen by the wraiths. When her magic power was stolen, her power continued to watch from inside the wraiths as Wraith-Madoka. Once Wraith-Madoka’s body fell apart, the magic power was freed and was able to return to Homura. While her magic power was inside the wraiths, it was able to meddle with a small part of them in order to go to Homura’s side and help guide her. However, she could not take total control over the wraith she overpowered and had to take drastic measures in order to return to Homura. The emotions that were stolen by the wraiths were used as a guide for Homura’s magical power to find a way to oppose the mutant wraiths. When Wraith-Madoka took Homura’s soul gem and Homura with her, she was able to temporarily acquire the rest of Homura’s powers in order to aid Kyoko and Mami in fighting off the rogue wraiths. After the body was destroyed, Homura’s magical power thought it wouldn’t be possible for the magical energy to return to her true body. While there were no guarantees it would work, there wasn’t any other way for her to proceed. Braided Homura asks what was in the shield that made the wraiths mutate and go rogue, and if it was that magic that caused this white-dressed Homura to come into existence. The white-dressed Homura, her magical power, explains that the origin of her time magic and her current magic differ from one another. In Homura’s soul gem there exists two kinds of magic, the shield of time manipulation and the bow of memory manipulation. White-dressed Homura causes the landscape to shift in order to better show how this second magic came to be. The two Homuras are transported to the same cosmic realm in which Homura bid farewell to Madoka. When Madoka’s wish changed the laws of the universe, Homura should have lost all memories of her. The flashback plays before their eyes as Madoka bids farewell to Homura and tells her it may be possible that she will return to her world with her memories of Madoka still, even if it’s a small chance a true miracle may still happen. Braided Homura asks why she managed to keep her memories of her even if the rest of the world forgot about her. White-dressed Homura responds that a true miracle did occur. When a magical girl makes a miracle happen, it gives birth to the crystallization of her soul . Homura and Madoka transcended the laws of the universe and their strong reciprocral feelings for each oher took form and became the memento that was passed from Madoka to Homura. From this the power to manipulate memories and the power to destroy wraiths was born, becoming Homura’s newfound powers. Braided Homura then asks if White-dressed Homura is the result of that wish and magic which she replies is correct. Because Homura’s second miracle was granted without a normal exchange, separated from the laws of the universe, it was considered an incomplete magic. Neither of the girls noticed this new miracle and the newfound power took temporary residence inside Homura’s soul gem. After some time, it began to interfere with Homura’s original magic which caused the shield, which was the path to an obsolete timeline, to suddenly link up with Madoka. Braided Homura looks uncertain and asks if what she saw inside the shield was no different than a witch. White-dressed Homura finishes her sentence for her and says that while she had no way to be sure if Madoka was indeed inside the shield fighting that witch, she just knew it to be true and had no choice but to erase her memories of the shield. Inside the vastness of the shield there lies a witch and wherever a witch exists then Madoka is sure to be there as well. But she chose to continue her mission with unshakeable conviction because she knew that someday, she would be freed from the burden that is the shield and be with Madoka once more. Forgetting about the shield in order to protect Madoka’s world became a miscalculation and when the shield was stolen by the wraiths, the magic of the shield got out of their hands and caused them to decompose. The innumerable curses contained inside came gushing forth and consumed the wraith that absorbed the magic and curses of the shield, causing it to consume the other wraiths around it until before long the uncontrolled curses transformed the bodies of the wraiths into that giant grotesque form as well as the mutant wraiths. The hopes and curses of humans are no something that the cause and effect laws of this world could assimiliate. Now the giant transformed wraith has begun its countdown on the annihilation of this world. Right now there isn’t a single wraith left that still carries its true purpose anymore, only the manifestation of a ‘true witch’ whose sole reason for existence is to bring calamity to the world. Should it not be stopped, this world will follow the same fate as the former world of witches. Braided Homura squeezes her fist and stands. She tells the white-dressed Homura if she really is her magic, to lend her power to take back the shield from the wraith’s hands before that thing destroys the world. Desperate, she tells her magic if she still has the power to manipulate time, she’ll rewind time until she can steal back her magic from the wraiths. Suddenly, Sayaka stands before her and asks her if she has the determination to kill that wraith since doing so may cause her to lose the time manipulation shield forever. Mami appears and warns Homura there's no guarantee her magic will return even if she does defeat the wraith. Kyoko also appears from nowhere and asks Homura if the shield is the proof that Madoka exists, didn't she doubt her existence just by tampering with her own memories. If she loses the shield, won't she suffer again the way she did when she doubted her own memories earlier, they ask. Sayaka steps forward, unsheathing her saber and taunts Homura, asking her if the reason she didn't destroy the shield when she realized Madoka could be inside it as well was because she wanted to meet with her. The three magical girls turn into the wraith versions of themselves and ask Homura if she wouldn't rather throw away this world and run away to the world of the shield where Madoka is. The Wraith-Madoka walks up to Homura and reminds her that she is a special power not bound by the laws of this world because she is a magical power born from her feelings for Madoka. Should Homura wish for it, a miracle could happen once more, even the power necessary to meet Madoka. Homura looks down, then decides she won't see her until she's defeated all the wraiths because it's her wish to protect Madoka. New powers or losing the shield won't change her wish and she couldn't let the world Madoka gave her life for be destroyed. Homura tried to run away from the pain of loneliness, and she still doubted the existence of the person who was more precious than anyone or anything and because of that people who had nothing to do with it were involved. But if she were to abandon this world and meet Madoka again, Homura doubts she would shower her with her smile. Wraith-Madoka holds Homura and tells her she did the best she could until the end. She looks at her and warns her she may doubt her memories again if she loses the shield, which is the thing that ties them together. Homura takes off her glasses and undoes her braids as she tells Wraith-Madoka that as long as the shield exists, she will never be able to meet Madoka again. Wraith-Madoka closes her eyes and becomes the White-dressed Homura once more as Homura turns to her and declares her words to be her feelings towards Madoka. She asks her magic to lend her power once more, not as the shield that protects Madoka but as the power that smites anyone who threatens Madoka. The Homura in the white dress glows and transforms into Homura's diamond-shaped soul gem, the same magic symbol that once adorned Wraith-Madoka in the center of the gem. The gem tells her it is the crystallization of her feelings for Madoka and to remember her precious unforgettable emotions for her. This is the miracle that was granted to Homura.

Back in the city a light glows brightly as the transformed giant-wraith's familiars hover in the city. Homura is suspended momentarily as she transforms into a magical girl, this time with her black bow in her hands and magic wings swirling from her back. She readies her bow for battle and a black rose sprouts from the end. She draws back an arrow as the familiars begin to swarm in her direction and throw black bolts of energy of their own. Homura unleashes a blinding arc of magic which completely disintegrates the hoard of familiars. As Homura prepares a giant crossbow of black swirling magic and aims it at the giant wraith, she realizes that the magic overflowing her body are her feelings for Madoka. She launches the arrow down straight at the core of the giant wraith which causes it to crack and burns the smaller wraiths on the ground. As Homura drives the arrow in further by hand, the wraith cracks open and the terrible eye of Kriemhild Gretchen stares back, the time shield where its pupil would be. Homura's soul gem tells her that this world and the previous world where she rewound time are different. Even if she were to activate the shield for example, it might not withstand the burden and she won't be able to rewind this horrible fate. Her soul gem asks if she's prepared for that. Homura passionately declares she's prepared to again and again make a miracle happen. She reaches for the time shield as in her head she asks Madoka to wait for her because one day she believes she'll come for her. But before she can reach it, several pins of black wraith magic pierce Homura's ribs in every direction, preventing her from reaching the shield. As she yells out in protest, her soul gem shines brightly and her magic wings flare back out. She uses them to push her even closer to the shield and strains to reach past the last few inches. Ultimate Madoka grasps Homura by the hand and tells her to give it her best, her silhouette reflected in Homura's eye. Homura touches the shield and it shatters, unleashing a blinding blast of light that envelopes the city and decimates the giant wraith.

Homura floats in an unknown realm where clock gears appear in the background. The White-dressed Homura is there and reaches her hand out to the real Homura. She explains how Madoka came to their help and thanks to her time is rewinding since the curses that shouldn't exist in this world was to change the fate of humanity. As the falling Homura lands in her giant hand, the White-Dressed Homura tells her that their memories will be erased as well this time but she will be here should Homura's resolve waver once more. The scene zooms out to show the White-dressed Homura crouching inside her soul gem.

Homura wakes up back at the station, where Mami and Kyoko stand looking saddened by Sayaka's unexpected departure. Madoka's ribbon remains in her hand. She holds the ribbon to her chest and says Madoka's name, but Mami and Kyoko can only ask who that is. Some time later, Homura sits on a construction rafter looking down on the city. Kyubey tells her that her story is nothing more than hypothesis with no way to prove it. Since Homura is the only one with memories of another world, to him it could be nothing more than a dream she had. Homura throws a handful of used up grief cubes at Kyubey as she watches the parade of wraiths walking the city streets. Kyubey warns her that the miasma is especially thick tonight with no end in sight of the wraiths she must destroy. Homura summons her bow and tells Kyubey not to complain as she drops down to battle the wraiths. Back up on the rafter, Kyubey pops the panel on his back open, revealing a fragment of something. It seems he is the only one to remember anything of the previous timeline of this world and he describes how touching the weapon or body of someone who manipulates time can cause you to share the same time as them. He reveals the fragment to be a piece of the broken time shield of Homura. Kyubey was able to personally witness the phenomenon of the law of cycles and remember everything thanks to this small sample of her weapon. Kyubey knows that one day her soul gem will fill with the color of curses and when that time comes Homura will surely cooperate with him for the sake of this universe. All for the sake of taking possession of the law of cycles, or as Homura called it Kaname Madoka. Homura walks into a desert filled with wraiths, her wings of black magic stretching out behind her as hundreds of Incubators watch from atop some nearby ruins.

[[Category:Wraith Arc Chapters]]

10^−43

2023-05-06T06:06:04Z

Agnesinφ: At the risk of overkill, added direct interwiki links to https://en.wikipedia.org/wiki/Template:Orders_of_magnitude and https://en.wikipedia.org/wiki/Order_of_magnitude, noting the wide-scale-spanning lists of quantities linked therein.

The '''Witch of Symbols''', presumably known as '''10^21''' ({{Runes|10}}^{{Runes|21}}) is a [[witch]] from the mobile game [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]]. Its existence was only known through its [[familiar]]s, the Echos, until the witch itself was revealed in the first chapter of Arc 2.5.

In the anime for Magia Record, a large Familiar with a green hue was seen within the Symbol Witch's barrier. Upon its defeat, the barrier disappeared. This is most likely the final form of the familiars as they grow through the different stages.

==Witch==
{{Card
|name = 10^21
|runes = 10 21
|image = File:1021.png
|japanese_desc = 象徴の魔女　その性質は『啓示』
抗えない使命を帯びているかのように、ひたすら繁殖を続ける魔女。意識を持たないので、出自や年齢はわからず、何をしたいのかも考えたことがない。ただ、自分を増やしたいという空疎な行動原理に従って、今まで増やしてきた自分の分身と連携しながら更なる分裂を続けている。しかし、繁殖以外の意思がないので、いくら数を増やしても、見つかると簡単に潰されてしまう。ゆえに魔女にとっての栄華はほど遠い。
|english_desc = A witch who simply continues to proliferate herself, as if charged with an inexorable duty. Because she has no consciousness, her place of origin and age are unknown, and she has never thought about what she wants to do. She simply obeys the hollow behavioural instinct to proliferate, working together with all of her previous offshoots to replicate herself ever further. But because she has no intent beyond proliferation, she’s easily stamped out when discovered, no matter how many of her there are. Thus, prosperity for this witch is still a long ways off.
|type = Symbol Witch
|nature = Revelation
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record]]
}}

== Minions ==
{{Card
|name = Echo
|runes = ECHO
|image = File:Card_Echo.png
|japanese_desc =
人間から奪った生命力に応じ成長する姿を変えると言われる象徴の魔女の手下達の中で、イス憑きと呼ばれる種類他魔女の手下に比べごく僅かな愉力を元に樣々な姿へと何段階も成長する。繁殖そのものを目的としており、ある程度成長した者は魔女にならずとも自身で新しい手下を産み落とすことが出来る。椅子を形どり、その利便性と完成されたフォルムを噛みしめて成長している模様。
|english_desc = Minions of the witch of symbols. Their role is to proliferate.
These Familiars of the Symbol Witch are called Haunting Chairs. They grow and transform by absorbing vital energy from humans. Compared to other Familiars they can change shape with a small amount of vital energy. Their primary goal is reproduction. Once they have grown enough, they can produce other Familiars, despite not being Witches. They grow, thoroughly enjoying the convenience and completeness of their forms.
|type = Symbol witch's minion I-IV
|task = Proliferate
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]], [[Magia Record Episode 6: I'll Do Anything|MagiReco Episode 6]]
}}
{{Card
|name = Echo
|runes = ECHO
|image =
|japanese_desc =
人間から奪った生命力に応じ成長する姿を変えると言われる象徴の魔女の手下達の中で、ヒト憑きと呼ばれる種類他魔女の手下に比べごく僅かな愉力を元に樣々な姿へと何段階も成長する。繁殖そのものを目的としており、ある程度成長した者は魔女にならずとも自身で新しい手下を産み落とすことが出来、1匹見かけたら30匹は存在すると恐れられる。見つけたら増える前に必ず駆除しよう。
|english_desc =
|type = Symbol witch's minion II-IV
|task = Proliferate
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]], [[Magia Record Episode 7: I'll Do Anything|MagiReco Episode 7]]
}}
{{Card
|name = Echo
|runes = ECHO
|image =
|japanese_desc =
人間から奪った生命力に応じ成長する姿を変えると言われる象徴の魔女の手下達の中で、トリ憑きと呼ばれる種類他魔女の手下に比べごく僅かな愉力を元に樣々な姿へと何段階も成長する。繁殖そのものを目的としており、ある程度成長した者は魔女にならずとも自身で新しい手下を産み落とすことが出来る。鳥を形どり、なんでも突き回しながら脳髄を揺らし、人類にはたどり着けない境地で成長している模様。
|english_desc =
|type = Symbol witch's minion II
|task = Proliferate
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]],
}}

==Trivia==
*The Witch was seen in [[Magia Record Story Puella Historia The Ikusamiko of Kamihama Arc|Puella Historia The Ikusamiko of Kamihama Arc]] during the Warring States Period. This could mean that the witch is around 500 years old.
*This witch's familiars and labyrinth have heavy feminine, fertility, and infertility symbolism.
** Her familiars grow from egg-shaped to being in cribs, then sitting in chairs (fertility), before finally becoming feminine humanoids with empty holes in their stomachs (infertility).
*** In the anime, the presumed final form of the familiars is humanoid figure standing on all fours, with an open hole where its head would be.
** The labyrinth features mannequins with pink broaches and rings (femininity), pink crystal pillars around which the rings sit and eggs rolling through the sky & across the ground (fertility), and dead, barren trees (infertility).
* The entrance to her labyrinth is a keyhole (femininity and fertility).
*Her name is presumed to be 10^''2''1. The third [[Runes|Rune]] corresponds to the [[Wikipedia:Caret|caret character (^)]]. The fourth Rune resembles that of the number 2, but is not an exact match (the exact Rune for 2, as it is known to look, [[Runes:Magia Record Season 2|was used in the second season]] of the [[Magia Record]] [[Magia Record Anime|anime adaptation]]). If it is indeed a new Rune, it is unclear what it corresponds to, but may be presumed to be a dash (-) or an underscore (_).
** 10^21 is the first name in the series that consists entirely out of numerals and punctuation marks.
***It is one of three mathematical expressions, or integers, based on the [[Wikipedia:Power of 10|power of 10]] (decimal [[wikipedia:Order_of_magnitude|orders of magnitude]]; see [[wikipedia:Template:Orders_of_magnitude|Template:Orders_of_magnitude]] for notable quantities of this and other sizes):
****10 to the power of 21 (or, the number 10 multiplied by itself 21 times). This would result in the number sextillion (1,000,000,000,000,000,000,000), whose name's prefix creates an English pun on reproduction.
****10 to the power of negative one (-1), resulting in 0.1.
****10 to the power of underscore one (_1), of unclear meaning. 10 to the power of just 1 results in 10.
***Her head is the planet [[Wikipedia:Mars|Mars]] (identifiable by the presence of the [[Wikipedia:Valles Marineris|Valles Marineris]] canyon), named for the Roman god of war and consort of Venus, goddess of love and sexuality.
***Her two halos (the rhombus/diamond above her head and the disc behind her), what appears to be a wing on her left side, and the large trumpet she wields, makes her resemble an Abrahamic angel.
***Her body resembles a block of marble stone with parts of it carved out as if to create a statue or piece of art.
****This, coupled with her name and the wording of her description, make her resemble a computer program or robot more than a living creature, and may indicate she is in some way an artificially created Witch.
*An [[Wikipedia:Echo|echo]] is the identical reflection of a sound off a surface, thus referencing that the Familiars are offshoots of the Witch that share her single-minded desire to reproduce into copies of her and each other.
**In Greek mythology, [[Wikipedia:Echo (mythology)|Echo]] was a nymph (a nature spirit) who was cursed for talking so much so as to only be able to repeat what others said. She later fell in love with a man named Narcisuss, who spurned her love and was thus himself cursed to fall in love with his reflection so intensely that he ultimately wasted away and died from not leaving his own sight. Eventually Echo faded until all that was left was her repeating voice.
**[[:File:Witch_178.PNG|Two of]] [[:File:Witch_177.PNG|the new]] Echo variants introduced in Arc 2.5 reuse [[:File:Episode_7_Familiar_Battle_1.png|designs]] [[:File:Episode_7_Familiar_Battle_2.png|seen in]] [[Magia Record Episode 7: I Want to Go Home with You|Episode 7]] of the game's anime adaptation. There, they were released from inside one of [[Alina Gray|Alina Gray's]] weapon containment cubes and had presumably been modified by her artistic endeavors and efforts to make them stronger.

==Gallery==
===Official Art===
<gallery>
File:Echo I - Magireco Archive.jpg|Echos from the Magia Archive
File:Echo II-IV - Magireco Archive.jpg
</gallery>
===Magia Record===
<gallery>
File:Witch_181.PNG|Witch bio
File:Enemy 6221 l.png|Witch sprite
File:Enemy 6223 l.png
File:Screenshot_2017-09-07-01-32-23.png|Echo I bio
File:Screenshot_2017-09-07-01-32-28.png|Echo II bio
File:Screenshot_2017-09-07-01-32-34.png|Echo III bio
File:Screenshot_2017-09-07-01-32-39.png|Echo IV bio
File:Enemy 7201 l.png|Echo I sprite
File:Enemy 7202 l.png|Echo II sprite
File:Enemy 7203 l.png|Echo III sprite
File:Enemy 7204 l.png|Echo IV sprite
File:Memoria 1014 c.png|Memoria card
File:Witch_177.PNG|New Echo variant seen in Arc 2.5
File:Witch_178.PNG
File:Witch_179.PNG
File:Witch_180.PNG
File:Enemy 7205 l.png
File:Enemy 7206 l.png
File:Enemy 7207 l.png
File:Enemy 7208 l.png
File:Symbol Witch 2nd.png
File:Symbol Witch 2nd name.PNG|Witch's 2nd name
File:Symbol Witch battle.PNG
File:Symbol Witch deferted.PNG|Witch 2nd defeated
</gallery>

====Barrier====
<gallery>
File:Symbol Witch's Barrier 1.jpg
File:Symbol witch barrier.jpg
</gallery>

===Anime===
<gallery>
File:Episode 6 Symbol Witch 1.png|Echos and their barrier from [[Magia Record Episode 6: I'll Do Anything|Episode 6]]
File:Episode 6 Symbol Witch 2.png
File:Episode 6 Symbol Witch 3.png
File:Episode 6 Symbol Witch 4.png|{{Runes|WE HAVE THE RIGHT TO BREED}} (We have the right to breed)
File:Episode 6 Symbol Witch 5.png
File:Episode 6 Symbol Witch 6.png
File:Episode 6 Symbol Witch 7.png
File:Episode 6 Symbol Witch 8.png
File:Episode 6 Symbol Witch 9.png
File:Episode 6 Symbol Witch 10.png
File:Episode 6 Symbol Witch 18.png
File:Episode 6 Symbol Witch 20.png
File:Episode 6 Symbol Witch 21.png
File:Episode 6 Symbol Witch 22.png
File:Episode 6 Symbol Witch 23.png
File:Episode 6 Symbol Witch 24.png
File:Episode 6 Symbol Witch 25.png
File:Episode 6 Symbol Witch 26.png
File:Episode 6 Symbol Witch 27.png
File:Episode 6 Symbol Witch 28.png
File:Episode 7 Alina's Barrier 9.png|Echos and their barrier from [[Magia Record Episode 7: I Want to Go Home with You|Episode 7]]
File:Episode 7 Alina's Barrier 11.png
File:Episode 7 Alina's Barrier 12.png
File:Episode 7 Alina's Barrier 13.png
File:Episode 7 Familiar Battle 1.png
File:Episode 7 Familiar Battle 2.png
File:Episode 7 Familiar Battle 5.png
File:Episode 7 Familiar Battle 6.png
File:Episode 7 Familiar Battle 25.png
File:Episode 7 Kyoko Rescue 7.png
File:Episode_7_Kyoko_Rescue_22.png
File:Episode 7 Kyoko Rescue 24.png
</gallery>

==External Links==
*[https://www.youtube.com/watch?v=-SOBYROa7jA Recording of 10^21's battle animations]

==Memoria Cards==
{{Magia Record Memoria Header}}
{{File:Memoria 1014 c.png}}
{{Magia Record Memoria Footer}}

==See Also==
*[[Puella Magi Madoka Magica Side Story: Magia Record]]
[[Category:Witches]]

10^−43

2023-05-06T05:55:45Z

Agnesinφ: Minor emendations. For any base n, n^1=n . Meanwhile 1 = n^0 = (n^a)/(n^a) = n^(a-a=0) for any base n and exponent a. Props for catching the pun, which had eluded me.

The '''Witch of Symbols''', presumably known as '''10^21''' ({{Runes|10}}^{{Runes|21}}) is a [[witch]] from the mobile game [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]]. Its existence was only known through its [[familiar]]s, the Echos, until the witch itself was revealed in the first chapter of Arc 2.5.

In the anime for Magia Record, a large Familiar with a green hue was seen within the Symbol Witch's barrier. Upon its defeat, the barrier disappeared. This is most likely the final form of the familiars as they grow through the different stages.

==Witch==
{{Card
|name = 10^21
|runes = 10 21
|image = File:1021.png
|japanese_desc = 象徴の魔女　その性質は『啓示』
抗えない使命を帯びているかのように、ひたすら繁殖を続ける魔女。意識を持たないので、出自や年齢はわからず、何をしたいのかも考えたことがない。ただ、自分を増やしたいという空疎な行動原理に従って、今まで増やしてきた自分の分身と連携しながら更なる分裂を続けている。しかし、繁殖以外の意思がないので、いくら数を増やしても、見つかると簡単に潰されてしまう。ゆえに魔女にとっての栄華はほど遠い。
|english_desc = A witch who simply continues to proliferate herself, as if charged with an inexorable duty. Because she has no consciousness, her place of origin and age are unknown, and she has never thought about what she wants to do. She simply obeys the hollow behavioural instinct to proliferate, working together with all of her previous offshoots to replicate herself ever further. But because she has no intent beyond proliferation, she’s easily stamped out when discovered, no matter how many of her there are. Thus, prosperity for this witch is still a long ways off.
|type = Symbol Witch
|nature = Revelation
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record]]
}}

== Minions ==
{{Card
|name = Echo
|runes = ECHO
|image = File:Card_Echo.png
|japanese_desc =
人間から奪った生命力に応じ成長する姿を変えると言われる象徴の魔女の手下達の中で、イス憑きと呼ばれる種類他魔女の手下に比べごく僅かな愉力を元に樣々な姿へと何段階も成長する。繁殖そのものを目的としており、ある程度成長した者は魔女にならずとも自身で新しい手下を産み落とすことが出来る。椅子を形どり、その利便性と完成されたフォルムを噛みしめて成長している模様。
|english_desc = Minions of the witch of symbols. Their role is to proliferate.
These Familiars of the Symbol Witch are called Haunting Chairs. They grow and transform by absorbing vital energy from humans. Compared to other Familiars they can change shape with a small amount of vital energy. Their primary goal is reproduction. Once they have grown enough, they can produce other Familiars, despite not being Witches. They grow, thoroughly enjoying the convenience and completeness of their forms.
|type = Symbol witch's minion I-IV
|task = Proliferate
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]], [[Magia Record Episode 6: I'll Do Anything|MagiReco Episode 6]]
}}
{{Card
|name = Echo
|runes = ECHO
|image =
|japanese_desc =
人間から奪った生命力に応じ成長する姿を変えると言われる象徴の魔女の手下達の中で、ヒト憑きと呼ばれる種類他魔女の手下に比べごく僅かな愉力を元に樣々な姿へと何段階も成長する。繁殖そのものを目的としており、ある程度成長した者は魔女にならずとも自身で新しい手下を産み落とすことが出来、1匹見かけたら30匹は存在すると恐れられる。見つけたら増える前に必ず駆除しよう。
|english_desc =
|type = Symbol witch's minion II-IV
|task = Proliferate
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]], [[Magia Record Episode 7: I'll Do Anything|MagiReco Episode 7]]
}}
{{Card
|name = Echo
|runes = ECHO
|image =
|japanese_desc =
人間から奪った生命力に応じ成長する姿を変えると言われる象徴の魔女の手下達の中で、トリ憑きと呼ばれる種類他魔女の手下に比べごく僅かな愉力を元に樣々な姿へと何段階も成長する。繁殖そのものを目的としており、ある程度成長した者は魔女にならずとも自身で新しい手下を産み落とすことが出来る。鳥を形どり、なんでも突き回しながら脳髄を揺らし、人類にはたどり着けない境地で成長している模様。
|english_desc =
|type = Symbol witch's minion II
|task = Proliferate
|episodes = [[Puella Magi Madoka Magica Side Story: Magia Record|Magia Record]],
}}

==Trivia==
*The Witch was seen in [[Magia Record Story Puella Historia The Ikusamiko of Kamihama Arc|Puella Historia The Ikusamiko of Kamihama Arc]] during the Warring States Period. This could mean that the witch is around 500 years old.
*This witch's familiars and labyrinth have heavy feminine, fertility, and infertility symbolism.
** Her familiars grow from egg-shaped to being in cribs, then sitting in chairs (fertility), before finally becoming feminine humanoids with empty holes in their stomachs (infertility).
*** In the anime, the presumed final form of the familiars is humanoid figure standing on all fours, with an open hole where its head would be.
** The labyrinth features mannequins with pink broaches and rings (femininity), pink crystal pillars around which the rings sit and eggs rolling through the sky & across the ground (fertility), and dead, barren trees (infertility).
* The entrance to her labyrinth is a keyhole (femininity and fertility).
*Her name is presumed to be 10^''2''1. The third [[Runes|Rune]] corresponds to the [[Wikipedia:Caret|caret character (^)]]. The fourth Rune resembles that of the number 2, but is not an exact match (the exact Rune for 2, as it is known to look, [[Runes:Magia Record Season 2|was used in the second season]] of the [[Magia Record]] [[Magia Record Anime|anime adaptation]]). If it is indeed a new Rune, it is unclear what it corresponds to, but may be presumed to be a dash (-) or an underscore (_).
** 10^21 is the first name in the series that consists entirely out of numerals and punctuation marks.
***It is one of three mathematical expressions, or integers, based on the [[Wikipedia:Power of 10|power of 10]]:
****10 to the power of 21 (or, the number 10 multiplied by itself 21 times). This would result in the number sextillion (1,000,000,000,000,000,000,000), whose name's prefix creates an English pun on reproduction.
****10 to the power of negative one (-1), resulting in 0.1.
****10 to the power of underscore one (_1), of unclear meaning. 10 to the power of just 1 results in 10.
***Her head is the planet [[Wikipedia:Mars|Mars]] (identifiable by the presence of the [[Wikipedia:Valles Marineris|Valles Marineris]] canyon), named for the Roman god of war and consort of Venus, goddess of love and sexuality.
***Her two halos (the rhombus/diamond above her head and the disc behind her), what appears to be a wing on her left side, and the large trumpet she wields, makes her resemble an Abrahamic angel.
***Her body resembles a block of marble stone with parts of it carved out as if to create a statue or piece of art.
****This, coupled with her name and the wording of her description, make her resemble a computer program or robot more than a living creature, and may indicate she is in some way an artificially created Witch.
*An [[Wikipedia:Echo|echo]] is the identical reflection of a sound off a surface, thus referencing that the Familiars are offshoots of the Witch that share her single-minded desire to reproduce into copies of her and each other.
**In Greek mythology, [[Wikipedia:Echo (mythology)|Echo]] was a nymph (a nature spirit) who was cursed for talking so much so as to only be able to repeat what others said. She later fell in love with a man named Narcisuss, who spurned her love and was thus himself cursed to fall in love with his reflection so intensely that he ultimately wasted away and died from not leaving his own sight. Eventually Echo faded until all that was left was her repeating voice.
**[[:File:Witch_178.PNG|Two of]] [[:File:Witch_177.PNG|the new]] Echo variants introduced in Arc 2.5 reuse [[:File:Episode_7_Familiar_Battle_1.png|designs]] [[:File:Episode_7_Familiar_Battle_2.png|seen in]] [[Magia Record Episode 7: I Want to Go Home with You|Episode 7]] of the game's anime adaptation. There, they were released from inside one of [[Alina Gray|Alina Gray's]] weapon containment cubes and had presumably been modified by her artistic endeavors and efforts to make them stronger.

==Gallery==
===Official Art===
<gallery>
File:Echo I - Magireco Archive.jpg|Echos from the Magia Archive
File:Echo II-IV - Magireco Archive.jpg
</gallery>
===Magia Record===
<gallery>
File:Witch_181.PNG|Witch bio
File:Enemy 6221 l.png|Witch sprite
File:Enemy 6223 l.png
File:Screenshot_2017-09-07-01-32-23.png|Echo I bio
File:Screenshot_2017-09-07-01-32-28.png|Echo II bio
File:Screenshot_2017-09-07-01-32-34.png|Echo III bio
File:Screenshot_2017-09-07-01-32-39.png|Echo IV bio
File:Enemy 7201 l.png|Echo I sprite
File:Enemy 7202 l.png|Echo II sprite
File:Enemy 7203 l.png|Echo III sprite
File:Enemy 7204 l.png|Echo IV sprite
File:Memoria 1014 c.png|Memoria card
File:Witch_177.PNG|New Echo variant seen in Arc 2.5
File:Witch_178.PNG
File:Witch_179.PNG
File:Witch_180.PNG
File:Enemy 7205 l.png
File:Enemy 7206 l.png
File:Enemy 7207 l.png
File:Enemy 7208 l.png
File:Symbol Witch 2nd.png
File:Symbol Witch 2nd name.PNG|Witch's 2nd name
File:Symbol Witch battle.PNG
File:Symbol Witch deferted.PNG|Witch 2nd defeated
</gallery>

====Barrier====
<gallery>
File:Symbol Witch's Barrier 1.jpg
File:Symbol witch barrier.jpg
</gallery>

===Anime===
<gallery>
File:Episode 6 Symbol Witch 1.png|Echos and their barrier from [[Magia Record Episode 6: I'll Do Anything|Episode 6]]
File:Episode 6 Symbol Witch 2.png
File:Episode 6 Symbol Witch 3.png
File:Episode 6 Symbol Witch 4.png|{{Runes|WE HAVE THE RIGHT TO BREED}} (We have the right to breed)
File:Episode 6 Symbol Witch 5.png
File:Episode 6 Symbol Witch 6.png
File:Episode 6 Symbol Witch 7.png
File:Episode 6 Symbol Witch 8.png
File:Episode 6 Symbol Witch 9.png
File:Episode 6 Symbol Witch 10.png
File:Episode 6 Symbol Witch 18.png
File:Episode 6 Symbol Witch 20.png
File:Episode 6 Symbol Witch 21.png
File:Episode 6 Symbol Witch 22.png
File:Episode 6 Symbol Witch 23.png
File:Episode 6 Symbol Witch 24.png
File:Episode 6 Symbol Witch 25.png
File:Episode 6 Symbol Witch 26.png
File:Episode 6 Symbol Witch 27.png
File:Episode 6 Symbol Witch 28.png
File:Episode 7 Alina's Barrier 9.png|Echos and their barrier from [[Magia Record Episode 7: I Want to Go Home with You|Episode 7]]
File:Episode 7 Alina's Barrier 11.png
File:Episode 7 Alina's Barrier 12.png
File:Episode 7 Alina's Barrier 13.png
File:Episode 7 Familiar Battle 1.png
File:Episode 7 Familiar Battle 2.png
File:Episode 7 Familiar Battle 5.png
File:Episode 7 Familiar Battle 6.png
File:Episode 7 Familiar Battle 25.png
File:Episode 7 Kyoko Rescue 7.png
File:Episode_7_Kyoko_Rescue_22.png
File:Episode 7 Kyoko Rescue 24.png
</gallery>

==External Links==
*[https://www.youtube.com/watch?v=-SOBYROa7jA Recording of 10^21's battle animations]

==Memoria Cards==
{{Magia Record Memoria Header}}
{{File:Memoria 1014 c.png}}
{{Magia Record Memoria Footer}}

==See Also==
*[[Puella Magi Madoka Magica Side Story: Magia Record]]
[[Category:Witches]]

User talk:AlsoArei

2021-11-29T02:41:03Z

Agnesinφ: Thanks for your cleanup.

Where did you find the extra info for some of the Mobage witches?! I'm intruiged!! :D [[User:DreamerWitches|DreamerWitches]] ([[User talk:DreamerWitches|talk]]) 09:52, 29 November 2020 (UTC)
The info on the giraffe witch comes from a comment by Kyubey on Mobage's main story. I assume the bird witch's is also from there, although I couldn't find it myself after some skimming. I found both on a-nima's blog and also a Japanese Madoka wiki. (Also, fun fact about the giraffe witch: lots of people speculate that her design is based around the Japanese idiom of "stretching your neck", meaning to wait eagerly. Like, she waited for so long that her neck got as long as a giraffe's.)

I've been wondering myself, where did you find the Madogatari images? I tried reverse google search but nothing showed up, not even your blog.

Also, good news, I found somebody willing to translate the Madogatari content! Stellar Canopy actually had thought of it already and had most of it done. I'm planning on adding it to the wiki over the next few days, but I thought I'd share a sneak peek: https://pastebin.com/sP0gqFgF
Feel free to edit it in youself if you want to. I still have to figure how to color text and add footnotes...--[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 21:11, 29 November 2020 (UTC)

Ohh! I see! That's great that we at least have some info on the Mobage witches!! It took a lot of time to find those images... I found it on this website https://www.4gamer.net/games/266/G026651/20160926088/ there's an image of a big wall of witch art and I had to zoom in on photoshop to find everything. And I'm super glad they're translating the stuff!!! ^^ Can't wait for it all to be done!!!! [[User:DreamerWitches|DreamerWitches]] ([[User talk:DreamerWitches|talk]]) 09:43, 30 November 2020 (UTC)

==Mobage page==
I'm super amazed by your work on the Madoka Magica Mobage page!! Thanks for all your additions! Though I was wondering if you would be able to find out during what events the Cell witch and Diva witch were added! Thanks! [[User:DreamerWitches|DreamerWitches]] ([[User talk:DreamerWitches|talk]]) 10:26, 22 December 2020 (UTC)

Okay, so I made a big post gathering evidence, which you can still check in the history... And then I realized none of it mattered, because I'd missed one crucial fact: There are seven more episodes than the wiki article says, and all the event witches appear there. All our screenshots of the fights are from this video, so there's no way to use them as a way to gauge when the witches were introduced.

I checked some old 4chan threads (or at least some Japanese site that seems similar) and while there are mentions of the frog and dream within the expected dates for their events (though apparently the frog witch's familairs, at least, were also out in the main story by then), but no such thing for the diva and cell witches. Perhaps they never appeared in events? But it's hard to prove it one way or another.

(Also, I started this post on a pretty bitter note, but thank you for appreciating my effort. I honestly wasn't sure if anybody cared about Mobage anymore)--[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 06:32, 23 December 2020 (UTC)

:Ah, I see, it's difficult then... gathering Mobage info is the hardest thing ever XD I really love Mobage, mostly for the witches but also cause there's little info on it and I want it to be uncovered! Thank you for your work!! ^^ [[User:DreamerWitches|DreamerWitches]] ([[User talk:DreamerWitches|talk]]) 10:25, 23 December 2020 (UTC)

== Soul gems ==
[[thumb]]

Thank you for adding information on the Soulgems page, but can you help me? In general, I have a lot of photos with soul gems, but little time for editing. In general, where can I contact you?
My talk page is fine. If I'm understanding you right, you want to give me the images and I edit them in? I don't mind the idea, but it depends on how much "a lot" is. Are you planning to add the soul gem of every Magia Record girl? Because I don't think the page can handle it (plus for those for whom we only know their transformed gems, we already note where they are on their pages so I don't think there's any need to add them). --[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 20:16, 18 January 2021 (UTC)
No no no, i meant Tart girls

Then sure. Feel free to post the images in my page and I'll add them later.--[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 20:53, 18 January 2021 (UTC)
Like here?
[[File:riz_gem_cropped.png|150px]]

Yes--[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 21:31, 18 January 2021 (UTC)

Ii have downloaded all the images i could find for the collabs from your google document, so that you dont have to download them yourself. ill begin to add them to my page. They were very useful links! :0)

== Alina Information ==

I don't think the information you removed from Alina's page should be removed. I think it's all noteworthy enough to remain, save for the point about her sanity. Her design does indeed resemble a prison warden just as Touka's and Nemu's designs also resemble prison elements. And of course Alina's other language wasn't Italian in the original dub, but clearly there had to be a choice by the ones in charge of the translations to make it Italian in the English dub instead of something like Latin, German, Spanish, Romanian, etc. - the franchise already had an established use of Italian with Mami, Kazumi Magica, and several other MagiReco characters. It's not hard to assume that's why they chose to use Italian where they did. --[[User:Garr9988|Garr9988]] ([[User talk:Garr9988|talk]]) 15:54, 30 September 2021 (UTC)

:I don't really see any resemblance to her design and a prison warden's, or Touka's and a prison. I know supposedly the resemblance is in Alina's hat and the metal parts of Touka's dress, but the latter exists in actual dresses and the hat is hardly specific to wardens. There's no official commentary suggesting so, either. In fact, according to Magia Archive, Touka is dressed in a goth loli style while Alina is supposed look like she's wearing a military uniform.

:As for the Italian, so far all instances of characters using Italian (the Kazumi cast, Kyoko, Madoka in a deleted CD scene. Not sure what Magireco characters you're refering to?) were explicitly tied to Mami in-universe, but there's no such relation for Alina. There are many reason why the translators might have chosen Italian. For example, it's an "artsy" language the reminds one of famous Italian artists like Michelangelo, but at the same time it shares romance language influences with English, allowing the use of words that look similar and mean the same thing. This would make it easier for English readers to understand her even without a translation. There could also be practical matters like if one of the staff members already knew some Italian or if there was a consultant previously picked out for any Mami lines, simplifying the matter of checking especifically Italian lines.

:Your suggestions don't give the same impression as Italian would, either. Latin is a dead language and considered very erudite, German is considered harsh and commanding, Spanish would be mostly associated with Mexico for USA players (and thus not quite as fancy and foreign) and I'm not sure Romanian is well known enough to have associations with anything except vampires (and even then most people probably wouldn't be able to recognize Romanian 'in the wild').

:In short, there's not enough evidence as to why Italian was chosen. Since originally Alina spoke in English, it's unlikely that any connection was intended in the first place. One might say that the translator were trying to create a previously-inexistent resemblance, but I think that's assuming too much unless there's an interview or something saying so. --[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 02:05, 1 October 2021 (UTC)

::Given that Nemu's design includes a ball-and-chain, it's not out of the realm of possibility that the other two Magius leaders' designs were also intended to evoke the feel of a prison. Just because there's no official commentary on it doesn't mean it isn't true, or at the very least that such a resemblance, if truly unintentional and coincidental, shouldn't be noted. That's what Trivia is for, is it not?

::As for the Italian in Magia Record, look to Iroha Tamaki's, Ui Tamaki's, Hinano Miyako's, Ria Ami's, and Yuuna Kaharu's (directly inspired by Holy Mami, but still) Magia attack names. I suppose there's something to be said about Magia Record's Italian (including Alina's localization) being potentially inspired by the Italian used in Kenji Miyazawa's "Night on the Galactic Railroad" instead of Mami, but still, the use of it is worth noting on the wiki either way. --[[User:Garr9988|Garr9988]] ([[User talk:Garr9988|talk]]) 15:31, 1 October 2021 (UTC)

Even the trivia section should be rooted 99% in fact. Otherwise it could be filled with just about anything, like "Sayaka's clothes are blue and white, therefore she's based on the sky. This ties her with Kriemhild Gretchen being a shadow on clouds. Sayaka also has a connection to Elsa Maria, who's black like the night sky. Furthermore, blueberries grow under the sky and Sayaka is compared to one in Rebellion". All of those facts are true, but we must be careful how far we stretch them, because things listed on a wiki page are automatically seen as trustworthy, and this can give the impression that statements are more definitive thatn they really are. This is especially important when there's relevant material unstranslated (like Magia Archive) that might lead people to think that it's stated there.

I agree that we should note the localization change, but I'd say it's an overreach to state why they did it. And if anything, the use of Italian by characters that have nothing to do with Mami makes it less likely that this is meant as a reference. Plus, several characters use English in their magias (off the top of my head, there's Oriko, Sasara, Shizuku and Chisato) but we don't consider them a reference to anyone in particular.

I think the idea of it coming from Galactic Railroad is interesting, but Magia Record likes to mix languages in general so it's hard to tell. Besides the already mentioned English and Italian, there's an effect named elpida (hope in greek), there are several magias in latin (mostly Madoka alts, but also Iroha-Yachiyo, and NY Madoka's magia also has German in it) and the Tart characters have magias or doppels in French.

Overall, I'd say that it's better to avoid information on a topic that to spread fanon, especially with how the puella wiki has taken a turn towards being more canon-focused in the last few years (you may notice that there's actually little arguable information in the newer trivia sections. For example, in Alina's page, the illustrator of her cards or her having a non-Japanese name aren't speculation).
If we prioritize the English version above the original one and start speculating on the translators' motives, we're observing canon from behind two layers of separation: the assumptions the translators brought to the text, and our own assumptions about what they were thinking. At that point, I think we're getting far enough from the creator's intentions that it doesn't really merit being mentioned in a wiki page. --[[User:AlsoArei|AlsoArei]] ([[User talk:AlsoArei|talk]]) 20:58, 1 October 2021 (UTC)

==Apologies for the sloppiness==
Thanks for catching my embarrassing accidental linepaste at the top of Livia's article. I do generally use a sandbox to reduce such errors but was yet unable to even create my own userpage per the policy here. Though checking now I see I could've created my own User_talk. =v.v= [[User:Agnesinφ|Agnesinφ]] ([[User talk:Agnesinφ|talk]]) 02:40, 29 November 2021 (UTC)

User talk:Agnesinφ

2021-11-29T02:18:09Z

Agnesinφ: Page creation test edit.

==Blueshift==
[[commons:File:Cosmos-animation_Lambda-CDM.gif|<math>W(1) = \Omega = \left(\int_{-\infty}^{\infty} \frac{dt}{\left(e^t-t\right)^2 + \pi^2}\right)^{-1} - 1\approx 0.567143290409783872999968662210355549753815787186512508135131079223045793086684566693219446961752294557638\ldots</math>]]

Mathematics of Madoka Magica

2021-11-22T23:47:58Z

Agnesinφ: Amended parentheses-position typo in Reb. B final line from "(x^2+y)\ln((x^2+y)-1)" to "(x^2+y)(\ln(x^2+y)-1)" to reflect that answer is of the form (x^2+y)\ln(x^2+y)-(x^2+y)+C , and not (x^2+y)\ln(x^2+y-1)+C.

[[File:Homura using statistics.jpg|thumb|[[Homura]] said that she uses applied statistics in witch hunts]]
The cast of Mahou Shoujo Madoka Magica are students of [[Mitakihara Middle School]]. Even though it is merely a middle school, coursework in Mitakihara can be quite challenging. The following are Math problems that have appeared thus far in the show, and their solutions:

==Episode 1==
[[file:Episode One Algebra.jpg|thumb|Episode one Math question 1]]

===Question 1===
Any integer divided by 14 will have a remainder between 0 and 13. Given that <math>a</math> has a remainder of 6 and <math>b</math> has a remainder of 1 when divided by 14, what is the remainder of <math>x</math> when divided by 14, given <math>x</math> is an integer solution to <math>x^2-2ax+b=0</math>?

'''Solution:'''

This problem can be solved simply with [[wikipedia:Modular arithmetic|modular arithmetic]]:

*<math>a \equiv 6 \pmod{14}</math>;
*<math>b \equiv 1 \pmod{14}</math>.

*<math>\displaystyle{x^2 - 2ax + b = 0}</math> implies

*<math>x^2 - 12x + 1 \equiv 0 \pmod{14}</math>

*<math>x^2 + 2x + 1 \equiv 0 \pmod{14} </math> (because <math>-12 \equiv 2 \pmod{14}</math>)

*<math>(x + 1)^2 \equiv 0 \pmod{14} </math>

*<math> x + 1 \equiv 0 \pmod{14} </math> (because 14 is square-free)

*<math> x \equiv 13 \pmod{14}</math> (because <math> -1 \equiv 13 \pmod{14}</math>).

'''Second Solution:'''

Homura used in Episode 1 a basic approach with usual integer [[wikipedia:arithmetic|arithmetic]].

Let <math>\displaystyle{x = 14q + r, a = 14s + 6, b = 14t + 1} </math>.

Substitute into <math> \displaystyle{x^2 - 2ax + b = 0} </math> to get after some calculations

<math>\displaystyle{x^2 - 2ax + b = 14c + (r+1)^2 = 0} \mbox{ with } \displaystyle{c = 14q^2 + 2q - 28qs - 2rs - 12q + t - r}</math>

14 divides <math>14c</math> and 0, hence it also divides <math>(r+1)^2</math>. This implies that <math>r+1</math> is divisible by 14.

The question asks for a remainder <math>r</math> between 0 and 13, so we obtain <math>r = 13</math>.

See also: http://archive.foolz.us/a/thread/44824340

===Question 2===
[[file:Episode One Math Q2.jpg|thumb|Episode one Math question 2]]
Assuming that <math>p</math> is a prime number and <math>n</math> is an arbitrary natural number, prove that <math>(1+n)^p - n^p - 1 \,</math> is divisible by <math>p</math>.

'''Solution:'''

By [[wikipedia:Fermat's Little Theorem|Fermat's Little Theorem]], for any prime <math>p</math> and integer <math>a</math>,

<math> a^p \equiv a \pmod{p} \,</math>

Thus:

<math>(1+n)^p - n^p - 1 \pmod{p} \,</math> is equivalent to

<math>(1+n) - n - 1 \pmod{p} \,</math> is equivalent to

<math>0 \pmod{p} \,</math>

So the overall expression is divisible by <math>p</math>.

'''Second solution:'''

The problem can be solved with the [[wikipedia:binomial theorem|binomial theorem]]:

For <math>a, b</math> not equal to <math>0</math> and nonnegative integer <math>p</math> it holds that:

<math>\displaystyle (a + b)^p = \sum_{k=0}^{p} \binom{p}{k} a^{p-k} b^k</math>

where the binomial coefficient <math>\binom{p}{k}</math> is the integer <math>\frac{p(p - 1) \cdots (p - k + 1)}{k(k-1) \cdots 1} </math> for <math>0 \leq k \leq p</math>.

Therefore,

<math>\displaystyle (1 + n)^p = \sum_{k=0}^{p} \binom{p}{k} n^k</math>

and

<math>\displaystyle (1 + n)^p - n^p - 1 = \sum_{k = 1}^{p - 1} \binom{p}{k} n^k</math>

since <math>\binom{p}{0} = \binom{p}{p} = 1</math>.

It holds that <math>\binom{p}{1} = p </math>.
Since <math>p</math> is a prime number, the factor <math>p</math> in <math>\binom{p}{k}</math> is not divisible by <math>k, k-1,\ldots, 2</math> for <math>2 \leq k \leq p-1</math>.
Therefore, <math>p</math> divides <math>\binom{p}{k}</math> for <math>1 \leq k \leq p-1</math>.

Since each summand on the right side is divisible by <math>p</math>, the whole sum, i.e. the left side is also divisible by <math>p</math>.

===Question 3===
[[file:Episode One Math Q3.jpg|thumb|Episode one Math question 3]]

Find the integer solutions <math>(a,b)</math> with <math>(a^3 + a^2 - 1) - (a - 1)b = 0 \,</math>.

'''Solution:'''
[[file:Episode One Math S3.jpg|thumb|Episode one Math solution to question 3]]

<math>
\begin{align}
(&a^3 + a^2 - 1) - (a - 1)b \\
& = (a^3 - 1) + (a^2 - 1) + 1 - (a - 1)b \\
& = (a - 1)(a^2 + a + 1) + (a - 1)(a + 1) + 1 - (a - 1)b \\
& = (a - 1)(a^2 + 2a - b + 2) + 1 = 0
\end{align}
</math>

Therefore, 
<math>(a - 1)(a^2 + 2a - b + 2) = -1 \,</math>

<math>a, b</math> are integers, therefore the factors <math>a - 1</math> and <math>a^2 + 2a - b + 2 \,</math> are integers too. Since the product is <math>-1</math>, one of the factors must be equal to <math>-1</math>, the other to <math>1</math>.

If <math>a - 1 = -1</math> then <math>a = 0</math> and from <math>a^2 + 2a - b + 2 = 1 \,</math> we obtain <math>b = 1</math>.

If <math>a - 1 = 1</math> then <math>a = 2</math> and <math>a^2 + 2a - b + 2 = -1 \,</math> implies that <math>b = 11</math>.

There are two integer solutions: <math>(a,b) = (0,1) \,</math> or <math>(a,b) = (2,11) \,</math>.

'''Second Solution'''

The question can also be solved by polynomial division.

<math>(a^3 + a^2 - 1) - (a - 1)b = 0 \,</math>

<math> b = \frac{a^3 + a^2 - 1}{a - 1} = a^2 + 2a + 2 + \frac{1}{a - 1} \,</math>

As <math>b \in \mathbb{Z}</math>, <math> a^2 + 2a + 2 + \frac{1}{a - 1} \in \mathbb{Z}</math>, so <math> \frac{1}{a - 1} \in \mathbb{Z}</math>

As <math> 1 </math> has only two factors: <math>1</math> and <math>-1</math>,

<math> a-1 = 1 \,</math> or <math> a-1 = -1 \,</math>

<math> a = 2 \,</math> or <math> a = 0 \,</math>

Substituting the above values into the equation to find the corresponding <math> b </math>, we have

<math> (a, b) = (0, 1) \,</math> or <math> (2, 11) </math>.

===Question 4===
[[file:Episode One Math Q4.jpg|thumb|Episode one Math question 4]]
Given <math>\displaystyle F(x) = \frac{4x + \sqrt{4x^2 - 1}}{\sqrt{2x + 1} + \sqrt{2x-1}}</math>,

find the sum of <math> \displaystyle{F(1)+F(2)+F(3)+...+F(60)}</math>.

'''Solution:'''
[[file:Episode One Math S4.jpg|thumb|Episode one Math, partial solution to question 4. Note that there are errors.]]

Simplying the fraction:

Notice that for any variables <math>a</math> and <math>b</math>:

<math> \displaystyle{(a-b)(a+b) = a^2-b^2} </math>.

Let <math> \displaystyle{a = \sqrt{2x + 1}} </math> and <math> \displaystyle{b = \sqrt{2x - 1}} </math>.

Multiply <math>\displaystyle{F(x)}</math> by 1 or <math>\displaystyle{\frac{a-b}{a-b}}</math> which is equal to <math> \displaystyle{\frac{\sqrt{2x + 1} - \sqrt{2x - 1}}{\sqrt{2x + 1} - \sqrt{2x - 1}}} </math>. The denominator becomes:

<math>
\begin{align}
&(\sqrt{2x + 1} + \sqrt{2x - 1})(\sqrt{2x + 1} - \sqrt{2x - 1})\\
&= (2x + 1) - (2x - 1)\\
&= 2
\end{align}
</math>

The numerator becomes:

<math>
\begin{align}
&(4x + \sqrt{4x^2 - 1} ) (\sqrt{2x + 1} - \sqrt{2x - 1})\\

&=(4x + \sqrt{(2x)^2 - 1)}) (a - b)\\

&=(4x + \sqrt{2x - 1} \sqrt{2x + 1} ) (a-b)\\

&=(4x + a b ) (a - b)\\

&=4x (a - b) + a^2 b - a b^2\\

&=4x (a - b) + (2x + 1) b - a (2x - 1)\\

&=4x (a - b) - 2x (a - b) + a + b\\

&=2x ( a - b) + a + b\\

&=(2x + 1) a - (2x - 1) b\\

&=a^3 - b^3

\end{align}

</math>

Therefore:

<math> \displaystyle {F(x) = \frac{a^3 - b^3 }{2}} </math>

Note that when <math> \displaystyle{ x = 1, 2, 3, 4, \ldots, 60} </math>;

<math> \displaystyle{ a^3 = 3^\frac{3}{2}, 5^\frac{3}{2}, 7^\frac{3}{2}, 9^\frac{3}{2}, \ldots, 119^\frac{3}{2}, 121^\frac{3}{2}} </math>;

<math> \displaystyle{ b^3 = 1^\frac{3}{2}, 3^\frac{3}{2}, 5^\frac{3}{2}, 7^\frac{3}{2}, 9^\frac{3}{2}, \ldots, 119^\frac{3}{2} } </math>;

Taking the sum over <math> \displaystyle{F(x)} </math> for <math> \displaystyle{x} </math> between <math>1</math> and <math>60</math>, observe that the majority of the terms in <math> \displaystyle{a^3} </math> and <math> \displaystyle{b^3} </math> cancel out, leaving:

<math>
\begin{align}
\sum_{x=1}^{60}F(x) &= \frac{121^\frac{3}{2} - 1^\frac{3}{2}}{2}\\
&=\frac{(11^2)^\frac{3}{2} - 1}{2}\\
&= 665
\end{align}
</math>

Thus, sum of <math> \displaystyle{F(x)} </math> for <math> \displaystyle{x} </math> between <math>1</math> and <math>60</math> is <math>665</math>. The solution can be generalized as equal to <math> \displaystyle{\frac{a(x_{\max}^3) - b(x_{\min}^3)}{2}} </math>. Where <math> \displaystyle{a} </math> and <math> \displaystyle{b} </math> are interpreted as functions of <math> \displaystyle{x} </math>.

==Episode 8==
Homura predicted the location of [[Walpurgis Night]] to be the clock tower using [[wikipedia:Statistics|statistics]]. The material Homura presents to Kyoko shows what looks like a [[wikipedia:Sinusoidal model|sinusoidal regression]], where a time series data is assumed to be following a sine curve over time and the properties (amplitude, frequency, phase) of the sine wave is predicted through statistics. For the purpose of predicting the location of [[Walpurgis Night]], statistics done this way would use geographic coordinates (longitude and latitude) as observed data points and attempt to fit two different sine waves to data, one for longitude and one for latitude. The fact that a large pendulum exists in Homura's room may be viewed as supporting this interpretation that what Homura used was a sinusoidal regression, because the mechanical movement of a pendulum also follows a sinusoidal pattern.

Irrespective of what type of statistics was used, the fact that [[Walpurgis Night]] actually appeared at the predicted location indicates that the model fitted the data very well with only a negligible amount of error.

==Episode 9==
[[File:Episode 9 Math.png|thumb|Fibonacci sequence question that appeared in Episode 9]]

This advanced [[wikipedia:Algebra|Algebra]] problem previously appeared in the first '''[[wikipedia:Tokyo University|Tokyo University]]''' Entrance Exam:

===Problem===
A number sequence <math> \displaystyle{ \{F(n)\} }</math> that can be defined as <math> \displaystyle{F(1) = 1, F(2) = 1}</math>, <math> \displaystyle{F(n+2) = F(n) + F(n+1)} </math> (where <math> \displaystyle{n \in \mathbb{N}} </math>) is called the [[wikipedia:Fibonacci number|Fibonacci sequence]] and its general solution is given by,

<math>\displaystyle{F(n) = \frac{\varphi^{n} - (-1/\varphi)^{n}}{\sqrt{5}}}</math>

<math>\displaystyle{\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887 \dots}</math>

Answer the following questions by using this fact if needed:

Define a sequence of natural numbers <math> \displaystyle{ \{X(n)\} } </math> (where n is any natural number), in which each digit is either 0 or 1, set by the following rules:

(i) <math> \displaystyle{X(1) = 1} </math>

(ii) We define <math> \displaystyle{X(n+1)} </math> as a natural number, which can be obtained by replacing the digits of <math> \displaystyle{X(n)} </math> with 1 if the digit is 0, and with 10 if the digit is 1.

For example, <math> \displaystyle{X(1) = 1, X(2) = 10, X(3) = 101, X(4) = 10110, X(5) = 10110101, \ldots} </math>

'' '''Anecdote''': <math> \displaystyle{X(n)} </math> can be considered a clever analogy to the show, where [[Episode 10]] should replaces [[Episode 1]] and [[Episode 1]] replaces Episode 0 in the viewer's next viewing.''

(1) Find <math> \displaystyle{A(n)} </math>, defined as the number of digits of <math> \displaystyle{X(n)} </math>.

(2) Find <math> \displaystyle{B(n)} </math>, defined as the numbers of times '01' appears in <math> \displaystyle{X(n)} </math>?
For example, <math> \displaystyle{B(1) = 0, B(2) = 0, B(3) = 1, B(4) = 1, B(5) = 3, \ldots} </math>

===Solution===
====Part 1====

Let <math> \displaystyle{A(n)} </math> equal to the number of digits in <math>X(n)</math>, which consists solely of 1s and 0s. Let's suppose <math> \displaystyle{x(n)} </math> is the number of 0s in <math> \displaystyle{X(n)} </math> at the n-th iteration (poor choice of variable by the student). Let's suppose <math> \displaystyle{y(n)} </math> is the number of 1s in <math> \displaystyle{X(n)} </math> in the n-th iteration, then <math> \displaystyle{x(n) + y(n) = A(n)} </math>. Since
*Every time a 0 appears, it is replaced with 1 at the next iteration, contributing to a single 1 in <math> \displaystyle{X(n+1)} </math>.
*Every time a 1 appears, it is replaced with 10 at the next iteration, contributing to a single 1 and a single 0 in <math>X(n+1)</math>.
it follows that the number of 0s in the next iteration is equal to the number of 1s previously:

<math> \displaystyle{x(n+1)= y(n)}</math> ;

and the number of 1s in the next iteration is equal to the number of 0s AND the numbers of 1s previously:

<math>y(n+1)= \displaystyle{x(n)+y(n)} </math> .

Next, prove that <math>x(n)</math> is a Fibonacci sequence, since we know that:

*<math> \displaystyle{y(n+1) = x(n) + y(n)} </math> ;
*<math> \displaystyle{x(n+2) = y(n+1)} </math> ;
*<math> \displaystyle{x(n+1) = y(n)} </math> ;

by substition we can show,

<math>
\begin{align}
&x(n+2)\\
&= y(n+1)\\
&= x(n) + y(n)\\
&= x(n) + x(n+1)
\end{align}
</math>

Hence, <math> \displaystyle{x(n+2) = x(n+1) + x(n)}</math>.

Thus <math> \displaystyle{x(n)} </math> fits the definition of a Fibonacci sequence. Since <math> \displaystyle{x(n)} </math> is a Fibonacci sequence, it follows that <math> \displaystyle{y(n)} </math> is also a Fibonacci sequence (given that <math> \displaystyle{x(n+1) = y(n)}</math>). Therefore, since it has already been shown that:

*<math> \displaystyle{x(n+2)=x(n+1)+x(n)} </math>

*<math> \displaystyle{y(n+2)=y(n+1)+y(n)} </math>

It follows that <math> \displaystyle {x(n+2)+y(n+2)=x(n+1)+y(n+1)+x(n)+y(n)} </math>.

Recall that <math> \displaystyle{A(n)=x(n)+y(n)} </math>, therefore

:<math> \displaystyle{A(n+2)=A(n+1)+A(n)} </math> .

Since, <math> \displaystyle{X(1) = 1, X(2) = 10, X(3) = 101, X(4) = 10110, X(5) = 10110101, \ldots} </math>

It follows that <math> \displaystyle{A(1) = 1, A(2) = 2, A(3) = 3, A(4) = 5, A(5) = 8, \ldots} </math>

Recall by definition of the Fibonacci sequence: <math> \displaystyle{\{F(n)\} = \{1, 1, 2, 3, 5, 8, \ldots\}} </math>. Therefore <math> \displaystyle{A(n) = F(n+1)} </math>, or

<math>\displaystyle{A(n) = F(n+1) = \frac{\varphi^{n+1} - (-1/\varphi)^{n+1}}{\sqrt{5}}}</math>

====Part 2====

Let <math>B(n)</math> be the number of times the sequence <math>01</math> appears in <math>X(n)</math>. Any two digits in <math>X(n)</math> may be <math>00</math>, <math>01</math>, <math>10</math>, <math>11</math>, and the corresponding digits in the next iteration <math>X(n+1)</math> will be

*<math> 00 \rightarrow 11</math>
*<math> 01 \rightarrow 110</math>
*<math> 10 \rightarrow 101</math>
*<math> 11 \rightarrow 1010</math>

Thus, any two digit sequence in <math>X(n)</math> that begins with 1* will contribute to a <math>01</math> sequence in the next iteration. In other words, <math>B(n+1)</math> equals <math>y(n)</math>, except when unit digit of <math>X(n)</math> is 1, or:

<math>B(n+1) = y(n) - odd(X(n))</math>

where
*<math>odd(n) = 1</math> if <math>n</math> is odd (unit digit is 1)
*<math>odd(n) = 0</math> if <math>n</math> is even (unit digit is 0)

It can be seen that <math>odd(n) = odd(X(n))</math> for all <math>n</math>. To prove this inductively: observe that <math>odd(X(1)) = odd(1)</math>. Now let <math>odd(X(n)) = odd(n)</math>, then if <math>n</math> is even, <math>X(n)</math> is even, thus <math>X(n)</math> ends with 0, which gets mapped to 1, making <math>X(n+1)</math> odd and so <math>odd(X(n+1)) = odd(n+1)</math>. And if <math>n</math> is odd, then <math>X(n)</math> is odd, thus <math>X(n)</math> ends with 1, which gets mapped to 10, making <math>X(n+1)</math> even, and so <math>odd(X(n+1)) = odd(n+1)</math>.

<math>y(n)</math> is a fibonacci sequence with starting values: <math>y(1) = 1</math>, <math>y(2) = 1</math>, <math>y(3) = 2</math>, thus <math>y(n) = F(n)</math>

<math>B(n+1) = F(n) - odd(n)</math>

<math>B(n) = F(n-1) - odd(n-1)</math>, or

<math>\displaystyle{B(n) = F(n-1) - odd(X(n-1)) = \frac{\varphi^{n-1} - (-1/\varphi)^{n-1}}{\sqrt{5}} - odd(X(n-1))}</math>

==Episode 10==
Same as [[Mathematics_of_Madoka_Magica#Episode_1|Episode 1]].
[[Category:Articles]]
[[Category:Fanmade analysis]]
[[Category:Science]]

==Episode 11==

[[File:Homura Walpurgis Ballistics Calculations.jpg|thumb|250px]]
Pieces of papers floats by Homura as she faces [[Walpurgis Night]]. On it are calculations done by hand by [[Homura]]. Most likely, these are [[wikipedia:ballistic|ballistics]] of the big guns. Two pairs of Xs and Ys on each of the four pieces of paper, making it 8 shots in all.

== Movie 3: Rebellion ==
[[File:Rebellion Math 1.jpg|thumb]]

=== Problem A ===

The first problem posed, (a), was to calculate <math>\int\frac{\arcsin^3x}{\sqrt{1-x^2}}\,dx</math>

We can perform the substitution <math>u=\arcsin x \longrightarrow \frac{du}{dx}=\frac{1}{\sqrt{1-x^2}}</math> and so rewrite the integral as <math>\int u^3\,du</math>

This can be easily solved via basic integration formulas: <math>\int u^3\,du=\frac{1}{4}u^4+C</math>

Undoing the substitution we get the solution: <math>\frac{1}{4}\arcsin^4x + C</math>

=== Problem B ===

The second problem, (b), was to calculate <math>\int x\ln(x^2+y)\,dx</math>
In order to solve it we'll assume <math>\displaystyle y</math> to be a parameter independent from the value of <math>\displaystyle x</math> and <math>\displaystyle\ln x</math> to denote the natural logarithm of <math>\displaystyle x</math>.

Performing the substitution <math>u=x^2+y\longrightarrow\frac{du}{dx}=2x</math> we can rewrite the integral as <math>\frac{1}{2}\int\ln u\,du</math>

This can be solved integrating by parts: <math>\frac{1}{2}\int \ln u\,du=\frac{1}{2}(u\ln u - \int1\,du)=\frac{1}{2}u(\ln u - 1)</math>

And the final answer in obtained undoing the substitution: <math>\frac{1}{2}(x^2+y)(\ln(x^2+y)-1)+C</math>

=== Problem C ===

One of the problems posed, (c), was to calculate <math>\int\frac{x^3+2x^2+10x}{x^2-x+1}\, dx</math>.

The answer is to first simplify the function:
<math>\frac{x^3+2x^2+10x}{x^2-x+1} = x + 3 + \frac{12x - 3}{x^2-x+1} = x + 3 + 6\frac{2x - 1}{x^2-x+1} + \frac{3}{x^2-x+1}</math>.
Note that <math>x^2-x+1</math> has only simple complex zeros.

It holds <math>\int x + 3 \, dx = \frac{1}{2}x^2 + 3x + C</math>.

The general formula <math>\int\frac{f'(x)}{f(x)}\, dx = \ln|f(x)| dx + C</math> can be used to calculate <math>\int \frac{2x - 1}{x^2-x+1} \, dx = \ln(x^2-x+1) + C</math>.

For the last term, <math>\int\frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan \frac{x}{a} + C</math> can be used (with some additional calculations) to obtain the solution

<math>\int\frac{x^3+2x^2+10x}{x^2-x+1}\, dx = \frac{1}{2}x^2 + 3x + 6\ln(x^2-x+1) + 2\sqrt{3} \arctan \frac{2x-1}{\sqrt{3}} + C</math>

Livia Medeiros

2021-11-21T03:02:22Z

Agnesinφ: Added links to en.wiktionary entries for easier verification. Evidence for a (presumably Buddhist) Sanskrit term "दुःख धातु"/"duḥkha dhātu" traditionally translated as "苦界" eludes me, but I know neither language well.

[[wiktionary:दुःख#Sanskrit|दुःख]] [[wiktionary:धातु#Sanskrit|धातु]]
{{Infobox_character
|name=Livia Medeiros
|image=[[File:Card 10414 l.png|300px]]
|jname= (リヴィア・メディロス) Livia Medeiros
|voices=
Japanese: [[wikipedia:ja:黒木ほの香|Honoka Kuroki]]
|id=1041
}}

'''Livia Medeiros''' is a character in the 2017 mobile game [[Puella Magi Madoka Magica Side Story: Magia Record]]. She first appeared in October 2019.

==General Info==
===Physical features===
* '''Age:''' ?
* '''Eye colour:''' Purple
* '''Hair colour:''' Purple

===Other===
* '''Soul Gem:''' Purple oval on her forehead
* '''Weapon:''' Top hat, playing cards (summoned from hat)
* '''Wish:''' Unknown
* '''Witch form:''' Luca
* '''Japanese pronoun:''' {{nihongo|watashi|私|}}
* '''School:''' -

==Game Info==
{|class="mw-collapsible mw-collapsed wikitable" style="width:100%;"
|-
!style="text-align:center; padding:10px;|Stats, Connect, Magia Data
|-
|style="padding:10px; background:#fff;"|

{|class="wikitable" style="width:100%;text-align:center;border:none;"
|style="width:25%;"|'''Rarity'''
|style="width:25%;"|'''Final Lv'''
|style="width:25%;"|'''Type'''
|style="width:25%;"|'''Attribute'''
|-
|style="width:25%;"|★4 → ★5
|style="width:25%;"|100
|style="width:25%;"|Support
|style="width:25%;"|[[File:attribute void.png]]
|-
|style="width:25%;"|'''STATS'''
|style="width:25%;"|'''HP'''
|style="width:25%;"|'''ATK'''
|style="width:25%;"|'''DEF'''
|-
|style="width:25%;"|'''Initial'''
|style="width:25%;"|6280
|style="width:25%;"|1809
|style="width:25%;"|1737
|-
|style="width:25%;"|'''Max at ★4'''
|style="width:25%;"|22281
|style="width:25%;"|6653
|style="width:25%;"|6117
|-
|style="width:25%;"|'''Max at ★5'''
|style="width:25%;"|28088
|style="width:25%;"|8400
|style="width:25%;"|7710
|-
|style="width:25%;"|'''Max SE'''
|style="width:25%;"|34083
|style="width:25%;"|10776
|style="width:25%;"|9833

|}
<onlyinclude>{|class="wikitable" style="width:100%;text-align:center;border:none;"
!colspan="5"|'''Disks'''
|-
|style="width:20%;"|[[File:disk accele.png]]
|style="width:20%;"|[[File:disk accele.png]]
|style="width:20%;"|[[File:disk blast horizontal.png]]
|style="width:20%;"|[[File:disk blast vertical.png]]
|style="width:20%;"|[[File:disk charge.png]]
|}
<onlyinclude>{|class="wikitable" style="width:100%;text-align:center;border:none;"
!colspan="5"|'''Connect: ''This is Just a One Time Thing'''''
|-
|style="width:100%;"|★4: Attack UP[VII] & Damage Up Versus Enemies Affected With Status Ailments[IV] & Guaranteed Curse
|-
|style="width:100%;"|★5: Attack UP[IX] & Damage Up Versus Enemies Affected With Status Ailments[VI] & Guaranteed Curse
|-
!colspan="5"|'''Magia: ''Epicurean's Game'''''
|-
|style="width:100%;"|★4: Damage to All Enemies [V] & Status Ailment Resistance DOWN & Defense DOWN (All / 3 T)
|-
|style="width:100%;"|★5: Damage to All Enemies [VII] & Status Ailment Resistance DOWN & Defense DOWN (All / 3 T)
|-
!colspan="5"|'''Doppel:''' ''Luca''
|-
|style="width:100%;"|Damage to All Enemies [IX] & Status Ailment Resistance DOWN & Defense DOWN (All / 3 T)
|}
|}

{|class="mw-collapsible mw-collapsed wikitable" style="width:100%;"
|-
!style="text-align:center; padding:10px;|Spirit Enhancement Data
|-
|style="padding:10px; background:#fff;"|

<onlyinclude>{|class="wikitable" style="width:100%;text-align:center;border:none;"
!colspan="7"|'''Max Stat Bonuses'''
|-
|style="width:10%;"|
|style="width:15%;"|'''HP'''
|style="width:15%;"|'''ATK'''
|style="width:15%;"|'''DEF'''
|style="width:15%;"|'''Accele'''
|style="width:15%;"|'''Blast'''
|style="width:15%;"|'''Charge'''
|-
|style="width:10%;"|4★
|style="width:15%;"|+8%
|style="width:15%;"|+5%
|style="width:15%;"|+5%
|style="width:15%;"|+6%
|style="width:15%;"|+6%
|style="width:15%;"|+4%
|-
|style="width:10%;"|5★
|style="width:15%;"|+9%
|style="width:15%;"|+6%
|style="width:15%;"|+6%
|style="width:15%;"|+7%
|style="width:15%;"|+7%
|style="width:15%;"|+5%
|-
|style="width:10%;"|SE
|style="width:15%;"|+9%
|style="width:15%;"|+6%
|style="width:15%;"|+6%
|style="width:15%;"|+13%
|style="width:15%;"|+13%
|style="width:15%;"|+10%
|}
<onlyinclude>{|class="wikitable" style="width:100%;text-align:center;border:none;"
!colspan="5"|Spirit Enhancement
|-
|style="width:100%;"|MP Gain UP When Over 100 MP [II]
|-
|style="width:100%;"|MP Gain UP When Over 100 MP [II]
|-
|style="width:100%;"|Doppel Damage UP [I] & Magia Damage UP [I]
|-
|style="width:100%;"|Doppel Damage UP [I] & Magia Damage UP [I]
|-
|style="width:100%;"|Guaranteed Anti-Curse
|-
|style="width:100%;"|Damage Cut [II]
|-
|style="width:100%;"|Blast Damage UP [II]
|-
|style="width:100%;"|Magia Damage UP [II]
|-
|style="width:100%;"|Damage UP Versus Enemies Affected With Status Ailments [II]
|-
|style="width:100%;"|Accele Damage Cut [II] & Magia Damage Cut [II]
|-
|style="width:100%;"|Chance to Curse on Attack [II]
|-
|style="width:100%;"|Chance to Skill Quicken [II]
|-
|style="width:100%;"|'''Active Ability:''' Chance to Curse [IV] & Defense DOWN [IV] (All / 1T) [''CD: 8 turns'']
|}
|}

==Summary==
A Magical Girl who has arrived in Kamihama as a [[Puella Care]] (Traveling Coordinator); she's referred to as Mitama's teacher. She has traveled around the country making adjustments and conveying her adjustment techniques to talented girls who can't fight witches on their own. She has a mysterious nature and no one knows her past.

===Doppel Description===
The Doppel of duḥkha-dhātu<ref>The word used is 苦界 (''kugai''), which is commonly read as as 'a life of prostitution' but can also be read as 'suffering world'. See [[wiktionary:दुःख#Sanskrit|दुःख]] and [[wiktionary:धातु#Sanskrit|धातु]].</ref>. Its form is a tree diagram.
Due to the presence of this being that represents the origin of life, the master of this emotion is painfully aware of the inescapable yoke that forces her closer to becoming one who does harm.
This Doppel speaks of nothing and indicates nothing, simply growing its tree of life ever larger with the impurity it absorbs from its master. Though its form is wrapped in swaddling cloth, it is every bit as unblessed as its master, and it harms those who oppose it by maliciously scattering the impurity it gained from its birth. If another being trapped within the tree diagram were to appear, they would surely fall into a crucible of agony. Only the master can become an actual part of the tree.
<references/>

===Side Story===

==Event Appearances==
*[[Magia Record Story Crimson Resolve|Crimson Resolve]]
*[[Magia Record Story Night Fairytale ～You Were At the Water's Edge～|Night Fairytale ~ You were at the water's edge]]

==Etymology==
* "Livia" as a name may either be derived from the Latin word ''liveo'' ("to be livid" or "to be envious"), or a variation of the name "Olivia" ("olive").
* The name Medeiros is derived from the Portuguese word, ''medeiro'', meaning "a place where stacks of maize are gathered", or simply "a haystack".
** The surname may come from Celestino Medeiros, a criminal who confessed to having committed a double-murder in ''Braintree'', Massachusetts for which [[wikipedia:Sacco_and_Vanzetti|two anarchist brothers]] were executed; Celestino's confession was ignored in favor of punishing the anarchists because there were strong political pressures at the time to execute them instead.

==Trivia==
* It has been confirmed that she is [[Mitama Yakumo]]'s sensei who taught her how to use her magic to adjust magical girls since she herself is an Adjuster as well.
* She is the first dark-skinned character in Magia Record.
* Her Doppel's name is a reference to the [[wikipedia:Last_universal_common_ancestor|Last Universal Common Ancestor]] and the fetuses represent, from left to right: mammals, reptiles, humankind, fish and birds.
** They also represent the four elements, with the mammal as earth, reptile as fire (lizards and salamanders are commonly associated with fire), fish as water and birds as air.
** It vaguely resembles a uterus without Fallopian tubes in shape. This could go along with the alternate reading of the kanji used for its nature.
** It also shares a name with one of the Souju twins from [[Puella Magi Kazumi Magica]].
* Her designer is Sasagi Koushi.

==Speculations==
{{Theory/Header|status=unknown}}
* Her personal Memoria could be interpreted to mean that Livia was cheated on due to that she pretend to not know until her marriage.
* Her personal Memoria and Doppel could also be interpreted to mean that Livia suffered a miscarriage which ended her marriage.
{{Theory/Footer}}

==Gallery==
<gallery>
File:card_10414_c.png|400px|Magia Record Four Star Art
File:tumblr_b6ef8c084ee828c1c67fd53c2dbe8731_eb8a9fea_1280.png|400px|Magia Record Five Star Art
File:Mini 104100 s.png|In-Battle Sprite
File:doppel_104100_l.png|Doppel Sprite
File:Livia doppel card.jpg
File:Unknown 10.png
File:Livia_Medeiros_Pre-Transformation.png
File:Livia Papa Seal 2021.jpg|Magical Girl Seal 2021
File:EOfhbHSUcAA-5Cq.jpg|400px|Damaged
File:Bg adv 14111.jpg
File:Bg adv 14121.jpg
File:Memoria 1447.jpg
File:Memoria 1449.jpg
File:Memoria 1450.jpg
File:102302 shizuka sleep.jpg
File:MagiReco OP 2 - 17.png
File:MagiReco OP 2 - 60.png
</gallery>

==Memoria Cards==
{|class="wikitable" style="width:100%;text-align:center;border:none;"
|style="width:10%;text-align:center;"|'''Card'''
|style="width:20%;text-align:center;"|'''Effect'''
|style="width:20%;text-align:center;"|'''Cooldown'''
|style="width:25%;text-align:center;"|'''Eng Text'''
|style="width:25%;text-align:center;"|'''Jap Text'''
|-
|style="width:10%;"|[[File:Memoria 1447 c.png|thumb|center|100px|'''''Blessing Unto You''''']]
|style="width:20%;"|'''Normal: ''' Damage CUT [III] & Magia Damage UP [IV] (Self / 3 T) '''Max Limit Break: ''' Damage CUT [IV] & Magia Damage UP [IV] (Self / 3 T)
|style="width:20%;"|'''Normal: ''' 7 turns '''Max Limit Break: ''' 6 turns
|style="width:25%;"|A chapel wrapped in voices of blessing.
Lift up the veil, swear a vow with those lips, toss out a cheer with the bouquet.
I thought I'd just feign ignorance, brush it off with a cold gaze,
but because I'd held such dear feelings, that love became warped.
|style="width:25%;"|祝福の声が包み込む教会
ベールをめくって、唇で誓いを立てて、ブーケでエールを送る
私は知らない振りをして、冷めた目で流すつもりだったのに、
大切に想ったから、愛は歪んでしまった。
|-
|style="width:10%;"|[[File:Memoria 1450 c.png|thumb|center|100px|'''''In Dreams are Sadness Permitted''''']]
|style="width:20%;"|'''Normal: ''' Regenerate HP [II] & MP Gain UP [II] (Self / 3 T) '''Max Limit Break: ''' Regenerate HP [III] & MP Gain UP [III] (Self / 3 T)
|style="width:20%;"|'''Normal: ''' 9 turns '''Max Limit Break: ''' 8 turns
|style="width:25%;"|Softly dozing, at the sound of adult voices that had once been so kind.
The innocent singing that remains makes the scene, and scent, feel that much more fresh.
If only it could've continued for 10 years, 20 years.
Once it crashed down, the foul-smelling time spent dragging it behind me seeped into the depths of my soul.
|style="width:25%;"|かつて優しかった大人の声を聞いて柔らかい微腫み。
残る無邪気な歌声が、情景と香りを鮮明にしていく。
10年も20年も続けば良かったのに。
崩れて引きずった臭い時間は、深く魂にまで染みついてしまった。
|-
|style="width:10%;"|[[File:Memoria 1472 c.png|thumb|center|100px|'''''Kindness Can Be Made''''']]
|style="width:20%;"|'''Normal: ''' Damage UP [IV] & Damage CUT [III] '''Max Limit Break: ''' Damage UP [V] & Damage CUT [IV]
|style="width:20%;"|Passive
|style="width:25%;"|"What's the matter, young lady? Have you lost your way?
Big sis will show you around. Follow me. ...well, that's the impression."
"Hmmm, I see...Instead of just telling me the direction of the place
it is a point of kindness to go together...! You have my gratitude, role model"
|style="width:25%;"|「お嬢ちゃん、どうしたん？　道に迷ってもうたんか？
お姉さんが案内したるわ。ついておいで。…とまあ、こんな感じや」
「むむ、なるほど…。ただ場所の方向を伝えるのではなく
一緒に行くのが優しさポイントなんですね…！お手本、感謝します」
|-
|style="width:10%;"|[[File:mitama_shadow_memoria.png|thumb|center|100px|'''''My True Nature is...''''']]
|style="width:20%;"|'''Normal: ''' Blast Damage DOWN [IX] (Single / 3 T) '''Max Limit Break: ''' Blast Damage DOWN [X] (Single / 3 T)
|style="width:20%;"|'''Normal: ''' 6 turns '''Max Limit Break: ''' 5 turns
|style="width:25%;"|No matter what I did, I couldn't fight. The only thing I could do was rely on others. I was rotting away, not knowing whether my wish was granted or not. Were those words whispered into my ear a blessing or a curse? "Mitama Yakumo, you are another Magical Girl who wished for a thousand curses."
|style="width:25%;"|何をしても戦えず…誰かに頼ることでしか生きられず…願いが叶ったかどうかも分からぬまま…朽ちようとしていた…その耳に囁きかけるのは、救いの言葉か呪いの言葉か…「八雲みたま、アナタもまた那由多の呪いを願った魔法少女だね」
|-
|style="width:10%;"|[[File:Memoria 1515 c.png|thumb|center|100px|'''''Rest Deliciously''''']]
|style="width:20%;"|'''Normal: ''' Damage Cut [III] & Defense UP While At Max Health [VII] '''Max Limit Break: ''' Damage Cut [IV] & Defense UP While At Max Health [IX]
|style="width:20%;"|Passive
|style="width:25%;"|“There are no customers, le’s all take a break together.”
“Very well, I shall roast the tea. I have also prepared the sweets. “
“Fumu-! Fummu-fumunmu!”
“Don’ go eatin' too much Sudachi. Ya won't have room for dinner after.”
|style="width:25%;"|「お客さんもおらんし、ここはみんなそろって休憩にしよか」
「では、お茶を煎れますね。お茶菓子も用意しましたので」
「ふむー！ふっむーふむんむ！」
「食べ過ぎたらあかんで月出里。ご飯食べれんくなるからな」
|-
|style="width:10%;"|[[File:Memoria 1449 c.png|thumb|center|100px|'''''The Traveling Adjusters''''']]
|style="width:20%;"|'''Normal: ''' Chance to Critical Hit [III] & Attack UP [I] & Anti-Charm '''Max Limit Break: ''' Chance to Critical Hit [IV] & Attack UP [II] & Anti-Charm
|style="width:20%;"|Passive
|style="width:25%;"|"If you wanna protect yourself from foes, us folks at Puella Care will lend you our strength.
But since we're absolutely neutral, we might Adjust your enemies too...
Well, if that happens, you can just drop by for another Adjustment yourself.
Might seem like a game of cat and mouse, but that's already the sorta place the world is."
|style="width:25%;"|「敵から身を護りたいんやったら、私らピュエラケアが力を授けたるで。
ただ絶対に中立である以上は、アンタの敵も調整するかもしれへん。
まぁ、その時はまた私らが調整して力をあげるわ。
イタチごっこやって思うやろうけど、そもそも世界がそんなもんやろ」
|}

==External links==
*[http://magireco.com Official Magia Record Website - Japanese]
*[https://magiarecord-en.com/ Official Magia Record Website - English]

===Gameplay Videos===
*[https://www.youtube.com/watch?v=I0WUuq40-Mo Recording of Livia's Transformation Video on Youtube (JP)]
*[https://www.youtube.com/watch?v=e6PZ7bcvCF0 Recording of Livia's Magia Video on Youtube (JP)]
*[https://www.youtube.com/watch?v=6BDOYiEdpbE Recording of Livia's Doppel Video on Youtube (JP)]

===Side Story and Costume Story Videos===
*Recording of Livia's MSS on Youtube (JP): [https://www.youtube.com/watch?v=P3Zqu4y-yFk Episode 1/3], [https://www.youtube.com/watch?v=FGt8X3dLWT4 Episode 2/3], [https://www.youtube.com/watch?v=fpLmOHTaiBo Episode 3/3]
**[https://youtu.be/_pbHsGHVwVs Recording of Livia's MSS on Youtube (JP) '''(Translated)''']

===Dialogue Lines===
*[https://www.youtube.com/watch?v=-BvAc7NaJvY Recording of Livia's Dialogue lines on Youtube (JP)]
**[https://pastebin.com/Jp5KeLp5 Pastebin with Livia's translated MSS and translated dialogue lines]

[[Category:Characters]]
[[Category:Magical girls]]