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The following are Mathematics problems that appeared in episodes of Madoka Magica and their solutions.
Notes from the breads for future pages:


[[file:Episode One Math Q2.jpg|thumb|Episode One Math Question 2]]
==madoka OP changes==
* = change
ep6:
green-grey trees
light blue veins, flows down wall
* turquoise-green -> purple/reddish purple -> red/yellow/green bg, light with white swirls sphere, yuri power hugging
swirling countdown
orange chainlink/roof/madoka/homura sequence + orange shaft of light
pink-orange moon title, light brown clouds
ED - swirling grey clouds/swirling blue clouds slow into a path


[[file:Episode One Math Q3.jpg|thumb|Episode One Math Question 3]]
ep7:
green-grey trees
light blue veins, flows down wall
* yellow-dark yellow bg, light sphere, gentle hugging
swirling countdown
orange chainlink/roof/madoka/homura sequence + orange shaft of light
pink-orange moon title, light brown clouds
ED - swirling grey clouds/swirling blue clouds slow into a path


[[file:Episode One Math Q4.jpg|thumb|Episode One Math Question 4]]
ep8:
==Episode One==
green-grey trees
light blue veins, flows down wall
yellow-dark yellow bg, light sphere, gentle hugging
swirling countdown
orange chainlink/roof/madoka/homura sequence + orange shaft of light
pink-orange moon title, light brown clouds
ED - swirling grey clouds/swirling blue clouds slow into a path


[[file:Episode One Algebra.jpg|thumb| Arithmetic question that appeared in episode one.]]
ep9:
The question reads:
green-grey trees
light blue veins, flows down wall
yellow-dark yellow bg, light sphere, gentle hugging
swirling countdown
orange chainlink/roof/madoka/homura sequence + orange shaft of light
pink-orange moon title, light brown clouds
* no ED


Any integer divided by 14 will have a remainder between 0 and 13.  Given a is has remainder of 6 and b has remainder of 1 when divided by 14, what is the remainder of x when divided by 14, given x is an integer solution to x^2-2ax+b=0?
ep10:
* OP at end instead of ED
* all text is black instead of pink and scrolls instead of fading in/out
green-grey trees
light blue veins, flows down wall
yellow-dark yellow bg, light sphere, gentle hugging
swirling countdown
orange chainlink/roof/madoka/homura sequence + orange shaft of light
* new tower top image with all 5 girls
pink-orange moon title, light brown clouds


'''Solution:'''
Your 'orange shaft of light' is just a lens flare.


This problem can be solved simply with modular arithmetrics:
==Mami sidescroller faq==
a = 6 mod 14;
from >>46592858
b = 1 mod 14;
You can ↓X dash into ↑X double jump to gain the most horizontal momentum possible. You can even jump clear over the first steps at the beginning of the level, even clearing the enemy's trail on Veteran.
x = r mod 14.
- ↓Z can be canceled into ↓X or aerial ←/→X. This is really useful for escaping Charlotte when she eats you. Conversely, you can also cancel dashes back into ↓Z, so you can go ↓X > ↓Z > →X for a really fast attack that covers a lot of ground.
- Kind of minor, but the fastest way to get access to small rifles is to do it on the ground and then shorthop up and catch them in the air. Aerial rifle spawn is obviously really slow, and this way is slightly faster than waiting for the grounded ones to land. Also, facing a wall and spawning rifles will spawn them all in the exact same place.
- Minor trick for Charlotte, if you go up on the top left platform of the three where she spawns, and stand all the way on the left, you can ↑Z and then move forward so that she spawns into a facefull of bullets.


x^2 - 2ax + b = 0 is equivalent to
Hold up and a to tiro finale


x^2 - 12x + 1 = 0 mod 14


x(x - 12) = -1 mod 14
==Magic & Abilities==


-1 has no proper divisors modulo 14, thus factors x and x-12 must correspond to either 1 or -1 mod 14.
Magic in Magica Universe are forces resulted by changes in emotions and an source energy that violates the First Law of Thermodynamics.  QB's civilization is a scientifically advanced alien race that is incapable of emotions, therefore incapable of utilizing magic for themselves.  When QB's civilization discovered magic, they scoured the Universe and found the human race, especially females at the second developmental period, to be most capable of developing such energy.  Thus they set out to develop a program to harnest this energy to delay heat death of the Universe.  Mahou Shoujo and Witches are the result of this program.


If x = 1 mod 14, then x-12 = 3 mod 14: false. If x = -1 mod 14, then x - 12 = 1 mod 14: true.
Kyouko:Spear, Barrier, (possibly something food related to her gem)
Mami:Guns, UberGun, Can fight with a dirty gem as well as someone with a clean, ribbon power.
Sayaka:Rapiers of mass proportions, uber healing.
Homura:Stop-time, Go back in time, but only for a specific time period, has a hidden dimension in her clock-thing that she keeps everything


Therefore, x = -1 mod 14 = 13 mod 14, i.e., the remainder r of x is 13.
in.  
m an abilityfag so I love analyzing techniques and such.


Mami
Weapon: One-Shot Rifles
Ability: Magical catch-all ribbons, baseball bat augmentation


'''Second Solution'''
Sayaka
Homura used in Episode 1 a basic approach with usual integer arithmetic.
Weapon: Infinite Rapiers
Ability: Incredible Regen


Let x = 14q + r, a = 14s + 6, b = 14t + 1.
Homura
Weapon: Shield (Captain America ftw)
Power(s): Time Manipulation, Hammerspace


Substitute into x^2 - 2ax + b = 0 to get after some calculations
Madoka
Weapon: Cake Bow
Power: ?????


x^2 - 2ax + b = 14c + (r+1)^2 = 0 with c = 14q^2 + 2q - 28qs - 2rs - 12q + t - r
Kyoko
Weapon: Changling Spear
Power: Red Barrier


14 divides 14c and 0, hence it also divides (r+1)^2. This implies that r+1 is divisible by 14.
46949443
Anything else that I'm missing?


The question asks for a remainder r between 0 and 13, so we obtain r = 13.
I'm thinking that each MG has an offensive, defensive, and supplementary power. only the supplementary power is affected by the wish.  


See also: http://green-oval.net/cgi-board.pl/a/thread/44824340
Here's how I see the breakdown:
Mami:
-offensive: rifles/muskets; defensive: the ribbons
-supplementary: magical longevity (based on her wish to live)


Sayaka:
-Offensive: swords; defensive: unknown (the main prob with the theory)
-Supplementary: incredible regen


==Episode Nine==
Homura:
-Offensive: hammerspace; defensive: shield (seen in the ep1 fight)
-Supplementary:time manipulation (based on wish to redo her meeting until able to protect Madoka)


[[File:Episode 9 Math.png|thumb|Fibonacci sequence question that appeared in Episode 9]]
Kyoko:
-Offensive: jointed spear; defensive: barrier
-Supplementary: ability to manipulate decisions (based on her wish to make people believe/follow her father)


This advanced Algebra problem previously appeared in the 2003 Tokyo entrance exam:
Madoka:
-Offensive: bow and arrow; defensive: not known (not much screen time)
-Supplementary: unknown (because wish unknown)


Change in Mahou Shoujo = Contract rate - (rate of Dead + rate of Becoming witch) * Current number of Mahou Shoujo;


The text reads:
A number sequence {F(n)} that can be defined as F(1) = 1, F(2) = 1, F(n+2) = F(n) + F(n+1) (where n is any natural number) is called the Fibonacci sequence and its general solution is given by,


[[file:fibonacci.png]]
==Population dynamics==
Change in Witches = Number of minions that mature per witch * current number of Witches - rate of Witches killed * current number of Mahou


[[file:golden ratio.png]]
Shoujo + rate of Becoming witch * current number of Mahou Shoujo
 
ΔM= C - (D + B) * M(t)
Answer the following questions by using this fact if needed:
ΔW= N * W(t) + (B - K) * M(t)
 
 
Define a sequence of natural numbers X(n) (where n is any natural number), in which each digit is either 0 or 1, set by the following rules:
 
(i) X(1) = 1
 
(ii) We define X(n+1) as a natural number, which can be gotten by replacing the digits of X(n) with 1 if the digit is 0, and with 10 if the digit is 1.
 
For example, X(1) = 1, X(2) = 10, X(3) = 101, X(4) = 10110, X(5) = 10110101, ...
 
 
(1) Find A(n), defined as the number of digits of X(n)?
 
(2) Find B(n),defined as the numbers of times '01' appears in X(n)?
For example, B(1) = 0, B(2) = 0, B(3) = 1, B(4) = 1, B(5) = 3,...
 
 
'''Solution'''
 
''Part (1)''
 
Let A(n) equal to the number of digits in X(n), which consists solely of 1s and 0s. Let's suppose x(n) is the number of 0s in X(n) at n iteration (poor choice of variable by the student). Let's suppose y(n) is the number of 1s in X(n) at n iteration.
 
 
1) Every time a 0 appears, it is replaced with 1 at the next iteration.
 
2) Every time a 1 appears, it is replaced with 10 at t he next iteration.
 
Therefore, the number of 0s in the next iteration is equal to the number of 1s previously:
 
x(n+1)= y(n)
 
and the number of 1s in the next iteration is equal to the number of 0s AND the numbers of 1s previously:
 
y(n+1)=x(n)+y(n)
 
 
Prove: x(n) is a fibonacci sequence:
 
i) y(n+1) = x(n) + y(n)
 
ii) x(n+2) = y(n+1)
 
iii) x(n+1) = y(n)
 
 
complete the substition shows:
 
x(n+2) = y(n+1) = x(n) + y(n) = x(n) + x(n+1)
 
x(n+2) = x(n+1) + x(n) ie definition of fibonacci sequence
 
Since x(n) is a fibonacci sequence, y(n) is also a fibonacci sequence since x(n+1) = y(n). Therefore:
 
x(n+2)=x(n+1)+x(n)
 
y(n+2)=y(n+1)+y(n)
 
x(n+2)+y(n+2)=x(n+1)+y(n+1)+x(n)+y(n)
 
Recall: A(n)=x(n)+y(n), therefore
 
A(n+2)=A(n+1)+A(n)
 
X(1) = 1, X(2) = 10, X(3) = 101, X(4) = 10110, X(5) = 10110101, ...
 
therefore: A(1) = 1, A(2) = 2, A(3) = 3, A(4) = 5, A(5) = 8,...
 
Recall fibonacci sequence F(n): 1, 1, 2, 3, 5, 8.
Therefore A(n) = F(n+1), or
 
[[file:An.png]]
 
 
''Part (2)''
 
Let B(n) be the number of times '01' appears in X(n).
 
Any two digits in X(n) may be 00, 01, 10, 11
 
the corresponding digits in the next iteration X(n+1) will be
 
00 -> 11
 
01 -> 110
 
10 -> 101
 
11 -> 1010
 
Thus, any two digit sequence in X(n) that begins with 1* will contribute to a 01 sequence in the next iteration. In other words, B(n+1) equals y(n), except when unit digit of X(n) is 1, or:
 
B(n+1) = y(n) - odd(X(n))
 
y(n) is a fibonacci sequence with starting values: y(1) = 1, y(2) = 1, y(3) = 2
thus y(n) = F(n)
 
B(n+1) = F(n) - odd(X(n))
 
B(n) = F(n-1) - odd(X(n-1)), or
 
[[file:Bn.png]]
 
== entropy post from general ==
 
notes on entropy from general
http://www.youtube.com/watch?v=TbRSd2THTUQ
31:00 Entropy explains why sea ...
 
“Assume Incubators are trying to sustain the entire universe using the misery of little girls. According to data gathered by the Wilkinson Microwave Anisotrophy Probe (WMAP), energy density of the universe comes down to roughly 9.9 x 10^-30 g/cm3, or 0,889767627 joules/km^3. The expansion rate of the universe, again as determined by WMAP observations is roughly 71.9 (km/s)/Mpc, ignoring other studies. The minimum radius of the universe, as determined by cosmic microwave background radiation data, is 12 gigaparsecs. Consequently, the universe gains at least 862800 kilometers of radius per second, which comes down to a volume gain of 1.48656647 × 10^54 kilometers cubed per second.
 
Thus, in order to maintain the energy density of the universe (admittedly, though, there are better ways to preserve life) 1.32269872 × 10^54 joules is required every second. Let us use high estimates and say that there is one magical girl for every 1,000 people. As of now, there are roughly 6,9 billion living people, so there are 6,900,000 magical girls across the world. Assume that every magical girl defeats a witch every two days, and hands over her Grief Seed to an Incubator for its energy to be collected. This comes down to roughly 40 witches being defeated every second. Therefore, if the energy density of the universe is to be maintained, every Grief Seed needs to have an energy output of at least 3.31249766 × 10^52 joules. What can you do with this sort of energy?
 
a. Supply the Earth’s energy needs for roughly 8,6 octillion years, a timespan 6,2802117 × 10^17 times the current age of the universe.
b. Create 3,68565071 × 10^35 kilograms of any matter (except cheese.) This comes down to about 184027 sun-sized stars, 61 billion Earth-sized planets or a single supermassive black hole with a radius of 547298,036 kilometers, roughly 1,4 times the distance between Earth and the Moon.
c. In a single year, enough energy is collected to create roughly 10 galaxies the size of our Milky Way.
d. Create a single explosion a few hundred thousand times stronger than any gamma ray burst we’ve observed so far. This would end most, if not all life (at least carbon-based life as we know it) pretty much everywhere from Earth to Andromeda galaxy.”
 
Personally, the only reason I can muster for this bizzare turn is that the writers want to change the setting. In the previous episodes, QB has become a bit too obviously antagonistic, making him a villain of sorts. But this development changes things, making QB not a villain but an act of god. Something outside human power. So the objective here becomes not “winning,” but rather “surviving”
 
==User talk==
:>Will there ever be a [[Mathematics and Madoka]] page?<br/>You can make one in the Speculah category (and link it from the [[Articles]] page) if you want.<br>As a side note, I think we should have a science page or something given all the discussion about lolentropy from this week. Maybe make a general page "Madoka and Science" and put all that stuff there in different sections? --[[User:Homerun-chan|Homerun-chan]] 15:10, 8 March 2011 (UTC)
::I am not good with wiki, so if you can make the Natural Sciences and Madoka category and a page out of this mess, I'll be hugely appreciative. [[User:Prima|Prima]] 00:27, 9 March 2011 (UTC)
 
I copypasted this page [[Madoka Magica and Science|here]]. You should edit that page from now on. At the time being, it is linked from the [[Articles]] page, but it may change in the future.<br>
Also, there is now a brand new "Science" category waiting to be filled. --[[User:Homerun-chan|Homerun-chan]] 17:14, 9 March 2011 (UTC)

Revision as of 03:22, 13 March 2011

Notes from the breads for future pages:

madoka OP changes

  • = change

ep6: green-grey trees light blue veins, flows down wall

  • turquoise-green -> purple/reddish purple -> red/yellow/green bg, light with white swirls sphere, yuri power hugging

swirling countdown orange chainlink/roof/madoka/homura sequence + orange shaft of light pink-orange moon title, light brown clouds ED - swirling grey clouds/swirling blue clouds slow into a path

ep7: green-grey trees light blue veins, flows down wall

  • yellow-dark yellow bg, light sphere, gentle hugging

swirling countdown orange chainlink/roof/madoka/homura sequence + orange shaft of light pink-orange moon title, light brown clouds ED - swirling grey clouds/swirling blue clouds slow into a path

ep8: green-grey trees light blue veins, flows down wall yellow-dark yellow bg, light sphere, gentle hugging swirling countdown orange chainlink/roof/madoka/homura sequence + orange shaft of light pink-orange moon title, light brown clouds ED - swirling grey clouds/swirling blue clouds slow into a path

ep9: green-grey trees light blue veins, flows down wall yellow-dark yellow bg, light sphere, gentle hugging swirling countdown orange chainlink/roof/madoka/homura sequence + orange shaft of light pink-orange moon title, light brown clouds

  • no ED

ep10:

  • OP at end instead of ED
  • all text is black instead of pink and scrolls instead of fading in/out

green-grey trees light blue veins, flows down wall yellow-dark yellow bg, light sphere, gentle hugging swirling countdown orange chainlink/roof/madoka/homura sequence + orange shaft of light

  • new tower top image with all 5 girls

pink-orange moon title, light brown clouds

Your 'orange shaft of light' is just a lens flare.

Mami sidescroller faq

from >>46592858 You can ↓X dash into ↑X double jump to gain the most horizontal momentum possible. You can even jump clear over the first steps at the beginning of the level, even clearing the enemy's trail on Veteran. - ↓Z can be canceled into ↓X or aerial ←/→X. This is really useful for escaping Charlotte when she eats you. Conversely, you can also cancel dashes back into ↓Z, so you can go ↓X > ↓Z > →X for a really fast attack that covers a lot of ground. - Kind of minor, but the fastest way to get access to small rifles is to do it on the ground and then shorthop up and catch them in the air. Aerial rifle spawn is obviously really slow, and this way is slightly faster than waiting for the grounded ones to land. Also, facing a wall and spawning rifles will spawn them all in the exact same place. - Minor trick for Charlotte, if you go up on the top left platform of the three where she spawns, and stand all the way on the left, you can ↑Z and then move forward so that she spawns into a facefull of bullets.

Hold up and a to tiro finale


Magic & Abilities

Magic in Magica Universe are forces resulted by changes in emotions and an source energy that violates the First Law of Thermodynamics. QB's civilization is a scientifically advanced alien race that is incapable of emotions, therefore incapable of utilizing magic for themselves. When QB's civilization discovered magic, they scoured the Universe and found the human race, especially females at the second developmental period, to be most capable of developing such energy. Thus they set out to develop a program to harnest this energy to delay heat death of the Universe. Mahou Shoujo and Witches are the result of this program.

Kyouko:Spear, Barrier, (possibly something food related to her gem) Mami:Guns, UberGun, Can fight with a dirty gem as well as someone with a clean, ribbon power. Sayaka:Rapiers of mass proportions, uber healing. Homura:Stop-time, Go back in time, but only for a specific time period, has a hidden dimension in her clock-thing that she keeps everything

in. m an abilityfag so I love analyzing techniques and such.

Mami Weapon: One-Shot Rifles Ability: Magical catch-all ribbons, baseball bat augmentation

Sayaka Weapon: Infinite Rapiers Ability: Incredible Regen

Homura Weapon: Shield (Captain America ftw) Power(s): Time Manipulation, Hammerspace

Madoka Weapon: Cake Bow Power: ?????

Kyoko Weapon: Changling Spear Power: Red Barrier

46949443 Anything else that I'm missing?

I'm thinking that each MG has an offensive, defensive, and supplementary power. only the supplementary power is affected by the wish.

Here's how I see the breakdown: Mami: -offensive: rifles/muskets; defensive: the ribbons -supplementary: magical longevity (based on her wish to live)

Sayaka: -Offensive: swords; defensive: unknown (the main prob with the theory) -Supplementary: incredible regen

Homura: -Offensive: hammerspace; defensive: shield (seen in the ep1 fight) -Supplementary:time manipulation (based on wish to redo her meeting until able to protect Madoka)

Kyoko: -Offensive: jointed spear; defensive: barrier -Supplementary: ability to manipulate decisions (based on her wish to make people believe/follow her father)

Madoka: -Offensive: bow and arrow; defensive: not known (not much screen time) -Supplementary: unknown (because wish unknown)

Change in Mahou Shoujo = Contract rate - (rate of Dead + rate of Becoming witch) * Current number of Mahou Shoujo;


Population dynamics

Change in Witches = Number of minions that mature per witch * current number of Witches - rate of Witches killed * current number of Mahou

Shoujo + rate of Becoming witch * current number of Mahou Shoujo ΔM= C - (D + B) * M(t) ΔW= N * W(t) + (B - K) * M(t)

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