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sinesalvatorem · 7 years

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Problem 6, Continued

OK, so, yesterday I said I was investigating Problem 6 of the 1988 International Math Olympiad. The problem as stated is:

Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that a2 + b2 / ab + 1 is the square of an integer.

I interpreted this to mean:

For any positive integers a and b such that ab + 1 divides a2 + b2, show that a2 + b2 / ab + 1 must be a perfect square.

In the process of investigating it, I would up proving that for any positive integers b and a such that b = a3, ab + 1 divides a2 + b2and a2 + b2 / ab + 1 is the square of an integer. This is related, but not quite what I’m looking for. My specific reasoning was:

Firstly, if a3 = b, and a and b are both squared in the numerator, then the numerator is effectively a2 + a6. Next, if the denominator is a*(a3) + 1, then it’s a4 + 1. Finally, the result of the equation is a2. Therefore:

a2 + a6 = (a4 + 1) * a2 #divide both sides by a2# 1 + a4 = a4 + 1

However, b = a3 isn’t the only time ab + 1 divides a2 + b2, nor the only time the result is a perfect square. For example, I noticed yesterday that a = 8 and b = 30 does it.

Anyway, that’s what happened yesterday, so now you’re all caught up. Now, the thing that nerdsniped me for the past two hours: What is the pattern of these “irregular” solutions, such as a=8 & b=30?

I fired up my compiler and wrote a script to search for pairs of numbers like this, and I discovered a pattern. The equation is solved when a=8 & b=30, or when a=30 & b=112, or when a=112 & b=418...

That is, there’s a cycle. The b of one solution is the a of the next, with some larger number being b next time. This goes on for as far up as I checked*, which means there’s a long (I expect infinite) sequence of integers where you can plug in any adjacent pair and find a solution.

However, this wasn’t the only sequence like this. In fact, after I stopped thinking of these as “the irregular solutions”, I realised there must be a bunch of sequences like this. Let me explain:

The sequence I just discussed ({8, 30, 112, 418...}) doesn’t start at 8. It starts at 2. Because, as you might recall, a=2 & b=8 is a solution. So we have a sequence {2, 8, 30, 112, 418...}.

(The pairs you can form have interesting properties - the first has a ratio of 4 (ie: 8/2 =4), while every subsequent pair has a slightly smaller ratio [3.75, 3.73333 (recurring), 3.7321 (truncated), 3.7320 (truncated)]. Because the difference between adjacent ratios becomes smaller as well, I assume this converges on some limit. However, at present, I’m not sure how I’d find it.)

However, if you’re like me, when you saw that 2 & 8 was the start of a sequence, you thought “Hold on, does that mean 3 & 27 is also the start of a sequence?” If so, you would be right! There’s a whole ‘nother “irregular” sequence {3, 27, 240, 2133, 18957...}. This also consists of pair-wise solutions, and the ratios also start at a perfect square (32) and then decline toward some (supposed) lower bound.

Unsurprisingly, there’s a sequence like this for every “regular” pair I checked (ie: pairs where b = a3), and I assume there’s one for all integer values of a. (Unproven, but conjectured strongly enough that I’d be pretty surprised if I were wrong.)

The result of the equation when given inputs from any of the sequences ends up being a square, of course. Specifically, a square of the first number in the sequence. So, for the first sequence I talked about, the equation always outputs 4. For this reason, I consider the first element of the sequence to be the defining element, so I’ve been calling each sequence Sfirst element. So, the first sequence discussed would be S2. (I expect there is a sequence S1, but that it simply contains {1, 1, 1, 1, 1, 1, 1...}).

The prime factorisations of the numbers in each sequence haven’t told me much useful stuff yet. So far, I can’t see a pattern that would let me predict** the next number in the sequence using its factorisation. Only two result really jump out at me:

The first thing is that every number in a given sequence has the first number of that sequence as a factor. This was intuitive to me so it wasn’t very interesting. What was interesting is that the third, sixth, ninth and so on elements of every single sequence tested are all divisible by 10. I have no idea why this keeps happening, but it does, and that’s cool enough that it might merit more research.

However, if anyone reading this discovers anything more interesting in the factorisations, please let me know.

Anyway, this has almost certainly all been a massive sidetrack from actually solving the problem. But, hey, I started working on the problem because it was interesting, and this sidetrack was interesting, so I don’t consider this lost in any way.

*I checked for the existence of all the sequences S2...S10, and several members of each set, and the factorisations of some members, and the ratios of some pairs. All those fiddly numbers and my obscure hypotheses are under the cut for anyone who thought this post wasn’t already long enough, or wants to use my results to investigate further.

**Edit: Shit, I only realised after writing this that I should just look these sequences up in the G-d damn OEIS. Sure enough, every sequence I’ve checked is already there, and has an equation for it. Here’s S2. I’m really kicking myself now. It’s a bit late to go diving into them, but I’ll let you guys know what I uncover.

Hi, extremely strange people who want to read even more math. Here is some more stuff from my exploration:

Firstly, here are my guesses about what might be going on here. Each conjecture has varying amounts of confidence, based on how intuitively correct the pattern seems to me and how confident I am that I understand the relevant math.

Conjectures with >95% confidence:

There are no natural numbers a such that b = a3does not yield a solution.

:. The set So is countably infinite.

:. There is a countably infinite number of sets S1...Sn.

All members of the set Sn will have n as a factor.

Conjectures with >70% confidence:

Each set S1...Sn is countably infinite

The set of all the sets S1..n (hence forth Sab) is countably infinite (for the same reason the rationals are countable)

Every third element (the third, the sixth, the ninth, etc) of any sequence S2...Sn will be divisible by 10

The limit for the ratios for a given sequence will be greater than n2-1, for any sequence Sn.

Conjectures with >50% confidence:

There are no solutions to the equation which are not in the super set Sab (bearing in mind that So is a subset of Sab., since the listing the first two elements of each set S1...Sn gives you the first 2n terms of So.)

For each set Sn, every prime is a factor of some element of that set. (Strictly conditional on all sets S1...Sn being infinite.)

If you can prove, disprove, or otherwise contribute to me thinking about any of these conjectures, you will win the prize of me expressing my gratitude by sending emoji to your inbox. That’s basically as valuable as a Field’s Medal, right? 💡💯💡

Next up, my method of investigation thus far:

The first thing I did was iteratively test loooooots of possible {a, b} pairs. Like, so many. Literally a million. Here’s the code that did that:

a = 5000 while((a > 0)): b = 200 while((b > 0)): if((a**2 + b**2)%(a*b + 1) == 0): x = (a**2 + b**2)/(a*b + 1) print("A =", a) print("B =", b) print("Numerator =", (a**2 + b**2)) print("Denominator =", (a*b + 1)) print("The result,",x, "is a square of", x**0.5) print(" ") b -=1 a -=1

This worked down from very high numbers to very low ones, testing lots of possibilities. The list generated included the original sequence of “regular” pairs (So, which contains {{1, 1}, {2, 8}, {3, 27}, {4, 64}...}. Unlike the Sn sequences, only the pairs shown as subsets are valid solutions.), where b = a3, up to {17, 4913}.

Additionally, it generated the first few terms of S2 and S3, as well as the first term of each of S4, and S5. This is the point at which it was 100% clear to me that there was a sequence of sequences S2..n. (I then assumed a sequence S1 which is as described before the cut.)

In testing the numbers in each sequence, I found that the ratios of the pairs declined as I described before the cut. The ratios for the first couple elements of S2 are:

8/2 = 4 30/8 = 3.75 112/30 = 3.73333 (repeating) 418/112 = 3.732142857 (truncated) 1560/418 = 3.732057416 (truncated) 5822/1560 = 3.732051282 (truncated) 21728/5822 = 3.732050841 (truncated) 81090/21728 = 3.732050810 (truncated) 302632/81090 = 3.732050808 (truncated)

As you can see, it appears to converge. I’m not sure how I’d figure out what the limit is, but I’m pretty confident it’s >3.7. Furthermore, you can see that each ratio is lower than the one before it, which means I could discount the possibility of a pair such that their ratio would be larger than the ratio of the pair before it. This allowed me to greatly restrict the search space when checking for each “family” of numbers (ie: numbers in the same sequence).

My method for searching within each “family” was to take its initial member ‘a’ (eg: In S2 that would be ‘2′) and then test whether a*a2 was a solution. Given that it was, I would enter it into my array, assign b to be that number, and then take the ratio of b/a to be the largest possible ratio of a pair of solutions.

Then I would use the number I’d just discovered as my new ‘a’, multiply it by the ratio I’d found last time, and then check if that number was a solution. If not, I’d decrement it by one over and over again until I found a solution, and then I’d save that and repeat the cycle, with the ratio becoming smaller each time.

This algorithm, converted to Python, was:

stopper = 1 ceiling = 25 family = [5] for a in family: b = int(a*ceiling) while(b > a): if((a**2 + b**2)%(a*b + 1) == 0): x = (a**2 + b**2)/(a*b + 1) ceiling = b/a family.append(b) stopper +=1 print("The ratio is", ceiling) break b -=1 if(stopper > 9): break print(family)

#This will print the first 10 elements of S5. If you want a different sequence Sn, then replace the ‘5′ in family’s array with n and the 25 for ceiling with n**2

Once the difference between the ratios of pairs became small enough, b ended up being pretty much the same as a*ceiling rounded down to the nearest integer.

OK, if you’re still reading this, now is when I paste all the numbers I found in case you want the values. First, I’ll list the first elements of the the sets I found, with the ratio of the last two elements in each list. I’ll give the first ten for S2 to S4, and the first 5 for the others up to S10. Underneath that, I’ll give you some prime factorisations, in case you’re into those.

S2: 2, 8, 30, 112, 418, 1560, 5822, 21728, 81090, 302632 302632 / 81090 = 3.732050808*

S3: 3, 27, 240, 2133, 18957, 168480, 1497363, 13307787, 118272720, 1051146693 1051146693 / 118272720 = 8.88748219*

S4: 4, 64, 1020, 16256, 259076, 4128960, 65804284, 1048739584, 16714029060, 266375725376 266375725376 / 16714029060 = 15.93725393*

S5: 5, 125, 3120, 77875, 1943755 1943755 / 77875 = 24.95993579*

S6: 6, 216, 7770, 279504, 10054374 10054374 / 279504 = 35.9722007556*

S7: 7, 343, 16800, 822857, 40303193 40303193 / 822857 = 48.9795833298*

S8: 8, 512, 32760, 2096128, 134119432 134119432 / 2096128 = 63.9843711834*

S9: 9, 729, 59040, 4781511, 387243351 387243351 / 4781511 = 80.9876524387*

S10: 10, 1000, 99990, 9998000, 999700010 999700010 / 9998000 = 99.9899989998*

(*All ratios are truncated)

Next, here are the prime factorisations for some of the elements of S2, S3, and S4:

2 = 2 8 = 2*2*2 30 = 2*3*5 112 = 2*2*2*2*7 418 = 2*11*19 1560 = 2*2*2*3*5*13 5822 = 2*41*71 21728 = 2*2*2*2*2*7*97 81090 = 2*3*3*5*17*53 302632 = 2*2*2*11*19*181

3 = 3 27 = 3, 3, 3 240 = 2, 2, 2, 2, 3, 5 2133 = 3, 3, 3, 79 18957 = 3, 71, 89 168480 = 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, 13 1497363 = 3, 7, 113, 631 13307787 = 3, 3, 3, 17, 79, 367 118272720 = 2, 2, 2, 2, 3, 5, 19, 37, 701

4 = 2 2 64 = 2, 2, 2, 2, 2, 2 1020 = 2, 2, 3, 5, 17 16256 = 2, 2, 2, 2, 2, 2, 2, 127 259076 = 2, 2, 239, 271 4128960 = 2, 2, 2, 2, 2, 2, 3, 5, 11, 17, 23

Hope that was helpful to anyone who actually checked!

#tragic backstory #math #problem 6 #long post cw

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captain-unlikelynutbouquet-blog · 6 years

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https://www.linkedin.com/pulse/medical-insurance-virginia-quotes-alexander-walters/"

0 notes

quonit-aceattorney · 6 years

Text

2-2 Reaction

Rules:

Q = Me, Quonit.

BF = Bardic Feline, the friend that made me spend 30 dollars on the game and whom I am messaging

I don’t use those when I send the messages close enough my username doesn’t appear.

Any typos (unless they are funny and part of the conversation) will be fixed.

Index

—

Q: also I can't play more ace attorney because when I finished game 1 my brother wanted to play it so now I can't play more until he realizes in order to play through game two he'll have to delete my progress so he has to wait till im done with that

BF: Whaaat

There should be more than one save slot!

Q :We thought that too! That ended up with Alex thinking the second game thing was another save file for the first game

BF: Huh

Q: he brought me back DS I can play as work yaaaay

BF: Wooo!

Q: back to phoenix wright

Q: car ooo

who was drugged with sleeping pills

oh hello ghost dude

murdurer who's that black haired girl outside of the flaming car

ini?

i liked these new opening scenes they have a lot more art

oh no D: not Maya!

they don't even avoid the "it was a rainy day" cliche like

who this guy

Q: uhhh turner that wasn't very friendly

Q: "I AM UPSET"

this was a year ago??? what???

Q: I was working under Mia not Maya

Q: the timeline makes no sense

June??? June what year??? Is this set before the chapter I just played???

oh my god that kid doesn't look right what

(^and that was my first ever reaction to Pearl. I thought she looked like a long 3-year-old. a glimpse. Ridicule me.)

Q: In happy Maya is here at least

Q: “I wouldn't take a crazed killer as my client" Maya you just have to BELIEVE they're not a crazed killer

Q: I wanna get to the case with Juan i n it but for now I'll be happy that at the moment he is living

Q: May 2nd is my bros birthday

Q: This guy needs better glasses that won't fall down

Q: Anyone can be a saint when compared to lawyers

Q: Wonder if anyone is in the direction

SIDE ROOM

Q: OH MY GOD THAT IS A LOT OF CANDLES

Is that a mushroom afro or hat

Q: I work hard at blaming it on Maya she gave me hints on what to do next and stopped the verdict from being assigned/mean any thing

Q: I wonder why we can't tell anyone about Mia

Q: Who is calling me!!!

HI I REMEMBER YOU

How are you lotta!

Q :How would Lotta plan to take pictures

Q: DID SOMEBODY DIE IN THERE

Oh now the citizens !Alex sense

CUT-SCENE

I think we should break in but Morgan won't approve

Q: Why will anybody who possesses Maya not stop showing off their books

Q: Boobs

Q: Hi gumshoe

Q: Who are you you don't look like you're in the Fea family

Q: Hey look it's the kid again

I got the key!! Thanks Pearl!

Q: Hey I don't have the key

Q: Ugh I'm stuck

BF: hahahah always that point where you end up trying to figure out what to do next

Oh, apropos of nothing, but Ini's name is pronounced "Eenie"

and she had a sister (who you will find out about) named Mimi

BF: and their last name is pronounced Mine-ee

Therefore, they are Ini (Eenie) Mimi (Meenie) Miney!

and the next case features a guy named MOE

Q: alright!

Q: Talking with the kid YAY

Reasons why I dislike the kindle

If shipping me and Maya gets me that key then please continue

Ya I wanna help in Maya

Q :This because brings up the question of if Maya likes me because everyone in the village listens to get and seems to ship us... Though I bet it's just a misinterpretation on their end

BF: Only on Pearl’s end, really

Q: As suspected

She's cute though! I like Pearl :D

Hopefully the aunt won't come and get mad at me

Q: Sorry to disappoint you pearl but I went to lawyer school not channeling school

BF: Pearl is such a cutie

Q: She is! I feel bad about the first thing I said about her is that she looks weird

Yesss finally I have the key

BF: Oh there’s going to be multiple things about Pearl’s reading skills, and before you judge her as a total idiot, remember that in the original Japanese, it’s Kanji she’s struggling with haha

Q: Alright!

Pearl what if I phrase it like this: MYSTIC MAYA told me to give this to you

Q: Okay pearl you can follow me as long as you know your mom won't put my head on a pike soon after

Q: How would gumshoe NOT like kids

Q: How about we explain to Gummy about how she was possessed

I do indeed have a copy

Gumshoe: The prosecutor is going to be Von Karma

WAIT WASN'T VAMPIRE ARRESTED

I SWEAR IF I ACTUALLY HAVE TO FIGHT THAT ASS AGAIN

Q: Oh thank God he didn't actually get out prison or something

Q: First time we see edgy in the game I wonder how he is doing

Q: PHOENIX somebody is actually interested in your badge and you don't show it to them???

How could you!!!

Q: IT'S NOT A HOUSE KEY, IT'S A BICYCLE KEY!

Q: It's that weird redheaded girl I don't like again

Q: The frick am I hallucinating?

Q: Oh hey she's lying!!! Thank you pearl

Q: Uh I don't know if I should try or not

Oh ya still stuck with this girl

Wait what

I looked at the second newspaper clipping and it says she died what

Q: Sharing a last name okay

Yay I win

Q: Ini if that is what happened why do you still look so happy

Q: :goes to court: I KNOW YOU ARE LYING BECAUSE OF THIS MAGIC ROCK AN 8 YEAR OLD HAVE ME

Q: Pearl let me take you away from your home without a parents consent I have some candy with me

Hi mia!!!

Q: So I know she can't tell me much because the game wouldn't be fun but what is the in universe explanation

Mia how dare you lie to me I thought we were friends

Save point

Q: Pearly hi how did you leave without your mom noticing

strong pearly

Edgy is my objection buddy

Wait he actually quit D:

I am ready to hate this person just as much as I did the first vampire

Q: No I am not shocked you are a woman now I'm just wondering why we have not met up with another female prosecutor

Q: HEY DON'T HIT THE JUDGE

Q: I don't think she should be allowed to bring that anymore

Then how did manfred fail if you are all so perfect

Fool

Q :HEY SHE HIT GUMMY D:

Q: Ya but we heard two gunshots and you say there was just one

Q: Hey there is a bullet hole in it

Q: I can figure it out I know I can

Q: I would use the gun on the last statement because it was fired twice and has the victims fingerprints on it but that didn't work

Q: Why can't I find anything

Q: I am stuck :(

BF: Aaaah I'd help but I don't know which part of the cross exam your at so I don't know what you're trying to prove right now

Q: I'll type the thing why not

Q: "Sorry pal there is more evidence", "this is the costume masthead wearing", "it's covered is blood," " defendant killed the person who did not right back ".

BF: ....I'm not sure, but I THINK you can find some evidence to disprove the 'didn't fight back' thing by gesturing to the sleeve of her costume somehow

If I'm remembering that segment correctly.

Q: Back

Hey the costume did work thank you

Q: Somebody please take that whip away from her

BF: hhaha no problem

the whip is Franzy's signature item, though. lD

BF: Also, she's not the last prosecutor who will physically assault you during a trial and get away with it hahah

Q: How dare this game

But there is a bullet hole in the outfit he didn't miss

Q: Finally that other bullet hole contradicts something

Q: I'm not getting something it's why I keep saving

I should know at this point where Maya was

BF: If you are still on the first day of the trial, you actually don’t have enough info yet to know where in the room she was!

Q: Dangit

Q: Oh hey I do know what I am doing

Q: I'm sorry what kinda swear is that Phoenix

Q: Lotta please don't be annoying like last time you were here

Oooo save point

Hi pearl I forgot about you

Q: See I'm not the only one annoyed by the whip!!!

How am I supposed to be happy when a Von Karma is around

How am I supposed to pretend we are all family

Q: Lotta there were two gunshots

Q: For now I don't think I can present any evidence

Q: Which is sad

Q: WHY DID YOU POSSES PEARL

at least your not showing off Boobs

Nevermind

Q: Von Karma I hate you

Q: FINALLY I can prove it wasn't Maya

Q: No morons that is Mia

Q: You know if was ready to hate this second vampire but she isn't nearly as bad as the first one so far

Q: I'm not a fool I'm just annoying

BF: Franziska really isn’t the straight up villain her dad was; she’s more like a more prickly Edgeworth

Q: Alright well that is better

I really wanna see pearl tell at her haha

Q: Ya edgy was fine after a while but being nice forced him to leave him job so

Q: Who is misty who are you trying to kill

Q: Not a movie but it is a game and anime so

Q: How did steel samurai merch get in here

Q: Aww a cute theme for pearl

Q: Don't worry pearl I won't tell your mom

Q: Hey Lotta if you don't want me to be chasing you then maybe you should stop being in the rooms I need to go to

Q: and that is why Phoenix doesn't date women Lotta.

Q: Lotta you're fine you're one of the more likable people in this game

Q: I dislike that girl so I'd be fine to say she's the killerI mean it'd make sense

She is the sister of girl who was killed by the man who just died

BF: Indeed, she does have a motive!

Q :back to thing

Comic is going it is fun to draw

Q: yes please give me info

new place to go. Also why does Lotta not like giving me info

what is this place

Q: you don't look like a doctor

Q: well of course you aren't

BF: Hahaha oh man I forgot you hadn’t encountered that creep yet

back to ace attourny but this is a great AU

Q: phoenix started talking like the guy hahaha

oh no did Pearl's mom call the cops on me

crap

Q: Ya well I suspected she was as evil

Q :need to go again dangit

Q: I think it's funny how I'm the scary one in this game

Q: I think right now I don't know if like the redhead girl or Morgan more

Q: the more I find out i guess i like the redhead more

Q: I win >:D

if we were both asleep then maybe you were both drugged like how the opening said

what is she talking about

im scared

Q: how about you show her your damn police badge that'd be interesting to her

pearl come with meeee

Q: mia don't lie to me i have a magic rock

Q: one more lock.

I feel smart :D

oh so morgan and ini were the killers great

Q: well if she made it look like Maya killed him she could become the next master

Q :Pearl did you run here AGAIN???

oh god please don't bring morgan

pfft Maya she does this to you too?

Ohhh well I guess that makes sense... I mean, she would be really mad at me for saying her mom killed the guy

Q: I like Mia and (whats her name the second vampire) arguing it's funny

Q: morgan you are lying

Q: ya no you hated me there is no way you thought "good sir"

BF: haha Franziska!

Q: I'll call her Franz for short

I mean this testimony was in the opening so maybe it isn't entirely a lie

Q: what do you mean it wasn't strange???

Q: but there was blood on her outfit how did you not think it was strange!!!

Q: mini stop eating your hat

that personality change reminds me of the worst person ever ... April May

Q: no but really it isn't more subjective

Q: Pearl is being used as evidence why not

Q: Franz stop it let me question her

i wanna present the urn but morgan is watching

but you know what there is nothing else here so

Franz: Why do I feel like you’re going to ruin my beautiful day? Me: CUZ I AAAAAMMMM

also why is Ini's hat so stretchy

BUT IT WAS A LIE

JUST BECAUSE SHE TESTIFIED ABOUT IT DOESN'T MEAN SHE CAN'T AGAIN

LIES AND SLANDER

Q :IT'S NOT BASELESS ALSO YOUR DAD IS A PIECE OF SHIT

Q: CRIME SCENE

SCENE OF THE CRIME HMMMM

Q: I am angry i will use caps

Q :YOU WERE HIDING THE PLACE PHOENIX ALWAYS THINKS SOMEBODY IS HIDING BUT WILL NEVER CHECK NO MATTER WHAT I DO

wow there are a lot of holes in your hat when you strech it out

spinny hat

then how did lotta earlier isn't she bigger than you

when i first went into the room the box wasn't there and then later it was

BUG ON MY DESK KILL IT

KILL IT NOW

i think it's dead

i hope so

Q :the folding screen >:3

SHUT UP FRANZ

ugh that bug is making me left arm flinch a lot

revenge after your death how dare you

Q :im not foolish im just better than you

BETTER THAN YOUR ASSHOLE OF A FATHER

I COULD RAISE YOU

wait what did i just say wtf hahaha

Q: and that is why i think that MORGAN HELPED

Q: WAIT I WANNA LOOK AT THAT FRAME BY FRAME

Q: yesss good case

well I mean your dad did

also fuck your dad I hate him that is all

BF: hahahaha

Q: actually no im not done

but finishing it would include a lot of swear words

BF: This case is totally setting up for some things in Game 3, btw

Q: dammit

Well I have motives for both of them Franz I just need to know that Phoenix knows.

BF: what you are ultimately trying to prove is Ini's motive, then that she did it and how she did it with Morgan's help

Q: okay!

Q: I am much more confidence than phoenix but that may just be anger

and a save point

OH GREAT NOW WE HAVE TO TALK TO FRANZ OUT OF THE COURT GREEAAAT

Actually Maya I had everything to do with it and I am proud I gave his ass the death penalty

stop being mean to Maya you're dad already cost Edgy his job hey now I have another reason to hate vampire

back to cort

Q: I'm sad because you said there is a universally hated case in the second game and I figured there are only 4 and the first two aren't it and neither is the fourth so it has to be the third but then I realized that Edgy is on the cover of the game so he should be in at least half of it so he might reappear in the worse one and I am sad

back to game

Q: LIAR YOU DID HAVE YOUR LICENSE THAT IS HOW THEY RECONSTRUCTED YOUR FACE

Q: “newb”

now this sounds like 2016

Q: how do you NOT know what the UK is you s***ty excuse for a judge

I'm so mad i started being mean to the judge I should direct it to Franz for being a jerk

BF: hahaha don't worry so much about that. I'm not going to say how it plays out, but I think you'll be pleased.

Q: yessss

BF: and the third case is the one that people generally don't like: I personally think it's interesting, but it DOES have some issues.

Q: dangit

well hopefully I can like it

BF: I'm going to let you make up your own mind on it, yeah.

Q: aaalllirght

BF: I'll tell you afterwards what I actually LIKE about it, okay?

Q: alright!

Q: WAIT SO

MINI IS PRETENDING TO BE HER SISTER HUH...

Q: aw hi mimi

Q: Fanz let me explain this to you: The Von Karma family is full of assholes you all deserve to die before they cause as much suffering as your father did

ya I'm going to call the cops on your for physical assult

are we allowed to tell Maya that Mia has been possesing her a lot

Q: game you know what the date and time is if you just don't want me to know tell me

So the games are just becoming more and more about magic huh

the end of the chapter this doesn't feel finished

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