# Baka to Tesuto to Syokanju:Volume7 The Fourth Question

## The Fourth Question

Use mathematical induction[1] to prove the following:

1 + 3 + 5 + … + (2n-1)=n2       ①

When n is a natural number.

1. When n=1, equation ① is calculated as,

Left Hand Side = 1
Right Hand Side = 1

2. Assuming that n=k is valid,

1 + 3 + 5 + … + (2k-1)=k2       ②

When n=k+1, the left hand side of ① is written as

1 + 3 + 5 + … + (2k-1)+(2k+1)
=k2 + (2k+1) (Obtained from equation ①)
=(k+1)2

In other words,

1 + 3 + 5 + … + (2k-1)+(2k+1)
=(k+1)2

① is valid when n=k+1.

From 1. and 2., equation ① is valid for all values on n.

Teacher's comment

That's correct. In mathematical induction, you need to prove that n=1 is valid. Also, when assuming that n=k is valid, you have to deduce that n=k+1 is valid. In this question, the key is that n is a natural number. On a side note, people often forget to put that n=1, so do take note when answering.

"I'll prove here that ① is correct - Tsuchiya Kouta"

Teacher's comment

Even if you try to sneak through by writing it in a style of a thesis, it's useless. The question mentioned that you have to prove it through 'mathematical induction', so while assuming that n=k, please write the equation that n=k+1 is valid.

"My judgment can hold."

Teacher's comment

“Oh! Akihisa, you're back.”

“Been tough on you, Akihisa.”

“...Welcome back.”

“Ah, mn, I'm back...”

As I finish the replenishment tests and return to the middle of the sports arena, Hideyoshi and the rest kindly receive my arrival. Seeing them like this at least alleviates the feeling of being left behind a bit. But whether we won or lost, I still wished to see it through with everyone...

Thinking about this, my eyes turn to the class F rest area that's bounded by rope. At this moment, I realize that the guys in class are surrounding a strange box and arguing. What are they doing?

“PLEASE LET ME BE THE LUCKY GUY!”

“STOP YAPPING AND START DRAWING! THERE ARE PEOPLE BEHIND!”

“I GOT IT, I GOT IT! WHY ARE YOU GUYS SO ANXIOUS! ALRIGHT, IT'S THIS ONE—DAMN IIIIIITTTTT!!!”

“Eh, what are they doing?”

I asked Yuuji who's beside me. Several classmates of mine seem to be dejected and unhappy. Why is that?

“Hm? You're talking about them? Just doing a lucky draw.”

“I know that. I just want to know why they're doing it.”

“Oh. The next event is the 2-people-3-legged race. The draw's to decide who goes with who.”

“Heh~so that's how it is.”

It's true that the choosing of partners in the 2-people-3-legged race is crucial. More than one's own ability, the most important thing is for both sides to be able to co-ordinate together.

“What's wrong, Akihisa? You're pretty calm.”

Hideyoshi asked me with a puzzled look, but there's no reason for me not to remain calm...

“Because it doesn't matter who's the partner. Guys and girls will be separated anyway--”

“This is a mixed event.”

“No problems at all. SUMMON!”

GO, MY SUMMONED BEAST! KILL THOSE GUYS FOR ME!!

“CALM DOWN, YOU IDIOT! DO YOU THINK YOU CAN SUMMON WITHOUT A TEACHER'S PERMISSION?”

“I CAN'T CALM DOWN AT ALLLL!! WHO IS IT? WHO'S THAT LUCKY GUY WHO GOT THE LUCKY NUMBER?”

“Relax, it's not decided yet.”

“Eh? Really?”

“...If that were the case, we wouldn't be quarreling like this.”

“Ah, you're right.”

Lucky...the key to winning the 2-people-3-legged race is to not get anxious and match the steps of the partner. The bodies will have to be together. If there's any traitor who would do such a great thing to the girls, that guy will be massacred by the FFF.

“I see. That's why everyone's praying.”

“That's the case.”

Everyone's praying before they stand in front of the box, putting their hands together and praying.

“But, to think that Himeji-san and Minami would agree to play the 2-people-3-legged race with the guys as well. Yuuji, did you say something to them?”

Looking over there, the other girls seem to be paired up with the same gender. Himeji-san and Minami probably did that because there are only 3 people, Himeji-san, Minami and Hideyoshi and can't leave someone out?

Hearing my words, Yuuji gives a 'really, you...' look.

“What idiotic words are you saying? I would wish for guys and girls to be in different groups.”

“Eh? Why—oh ya, I understand. You'll be killed if something goes wrong.”

“Good that you can understand.”

Once Kirishima-san knows that Yuuji's partner is a girl, no matter how far Yuuji is, she would install some counter measures. There's no such thing as 'impossible' in a girl-in-love's dictionary—everyone knows that.

“More than that, I feel that the baseball tournament's more important.”

“...I agree.”

Yuuji and Muttsurini nod their heads. These people have already turned into Asura just to get their confiscated A-books. They would prefer the more practical honor of getting back their A-books 10,000 times more than getting honor and glory.

“If that's the case, the next match is decided, right?”

The middle (first) sports arena is used to hold the sports meet, and the second arena that's located behind the school compound was used to hold our second-year baseball tournament.. The gym is opened for the 3rd years. We're facing the 3rd years later...

“...They haven't finished yet. It's extra innings.”

“Oh, if we're successful, we get a walk over.”

Due to the time limit, this match will be extended to 7 innings. If they can't win, the match will be decided as a tie. It sounds nice, but both sides can't progress on. If that's really the case, we can take the next match without playing.

“Anyway, let's check the next match to game plan...let me see, the next match has maths, biology, modern literature, economics and geography.”

Yuuji took out the schedule to check. These are all basic, normal subjects.

“...No health education.”

Muttsurini sounded rather lonely when he said that as he glanced at the schedule in Yuuji's hand. Like what Muttsurini says, there's no health education for every match.

“If that's the case, do you want to take part in the sports meet event, Muttsurini?”

“You're right. We'll do that for the second round.”

“...I got it.”

To think that Muttsurini would actually leave the frontlines. However, since there's no health education in the next few matches, it can't be helped. Besides, Muttsurini's other subjects are worse than me.

“Then we should readjust the batting order and the defensive alignments. The first one is maths, so Shimada will be the 1st batter...we'll let Sugawa handle the second. That guy seems like he can really bunt.”

“Then what about me? I can't handle maths.”

“You can't handle maths 'too'. Besides, even if we leave you at the back few batters, your biology scores are atrocious, so I might as well just leave you at 3rd. It's good if you can really bunt. If you're successful, we should be able to get 1 point.”

“Mn, got it.”

Yuuji decided our batting order and defensive positioning.

Just when Yuuji and I are discussing--

“Well, erm, Akihisa-kun...”

A voice called me from behind. This voice...Himeji-san?

“Hm? Himeji-san, what's wrong?”

Turning around, I see that Himeji-san and Minami are standing behind me. They're looking really serious. What's going on?

“It's not really something big...”

“Eh? You're asking me for my number...”

Ah, should be the batting order, right? As expected of Himeji-san, being so serious when she does things. She would actually check the defensive position and batting order. Though she's not really good at understanding sports, it seems like he's seriously thinking about this match.

Uu, my batting order...

“I'm 3. Number three.”

““WAH!!””

After I answered that, Himeji-san and Minami seem to whimper for some reason. Is my batting order weird?

“Uuu...I know the chances are slim, but I thought that it could be possible...”

“That's too much. This is too saddening...the one chance to stick on. This is a rare chance...”

Both of them are holding onto a little piece of paper each as they becried.

“AH~so you're talking about that. Don't worry, you two, Akihisa was just talking about the batting order. He hasn't drawn the number for the 2-persons-3-legged race.”

Yuuji said to them while sounding like he's lecturing them.

““Eh?””

Himeji-san and Minami, who were in a moment of melancholy just a second ago, lifted their heads up at the same time. Hm? Was the number they were talking about regarding the 2-persons-3-legged race?

“Oh ya. I forgot all about that. Got to take one number!”

Who can take it if those guys take the wonderful opportunity to be with the girls!?

“Ah, that...Akihisa-kun!”

“Wait, Aki!”

As I was about to go for the box of numbers, Himeji-san and Minami suddenly stopped me. Uu...if I don't hurry up, my happiness will be taken away.

“What? Is there something?”

“No. We...well...what should I say...I'm 7...”

“My number's 6...”

Himeji-san and Minami seem to say these numbers gingerly. 7 and 6? I see. So I'll strike jackpot if I get these numbers, right--

Seems like they really hate me.

“I, I got it...then, I'll draw one number then...”

“I say, you two. Those words may have already caused Akihisa to misunderstand...”

“Be, because it's Akihisa-kun...if we say to him 'please draw this number'--”

“That, that's right! When that happens, it will become the opposite of what we wish for. For some reason, I keep feeling that he will get Sakamoto's number or something...”

“...Really, it's been tough on you two...”

Sigh, though I never had much luck, I can't possibly get a good draw anyway. But I'm really sad to be told them by them...I drag my heavy footsteps and arrive in front of Sugawa-san, who's watching over the box.

“Uu, okay...”

With Sugawa-san staring at me mysterious, I reach my hand into the box. It'll be heaven if I can draw 6 or 7, so I have to do my best. Uu...okay, this one!

I immediately pull out the number. The number written on the paper is--

“Ah, 6...”

“KILL HIM!!!”

“““YES, HIEROPHANT!!!”””

“WHAT'S GOING ON? I GOT SURROUNDED!”

I haven't even said my number out loud, and a large group of people in black hoods surrounded me. I was careless! Saying out my number so openly like that, is it the end of me...

“6, 6...I see, so Aki will be with me...”

“Uu...Minami-chan looks so happy...”

“Do, do I? When do I look happy...”

“You liar! I can see your face glowing...”

“Uu...”

“I think Minami-chan will use this chance to touch Akihisa-kun's chest and butt...that's too sneaky...”

“Wh, what are you saying, Mizuki! How can I do such a thing—eh, what? What did you just say? Touching...Mizuki, what were you talking about...”

“Ah! No, nothing! I'm not talking about touching him sneakily. Th, that...I'm just talking about improving the relationship with Akihisa-kun. I just said it wrongly!”

“Mizuki..what were you planning to do to Aki...”

In a blink of an eye, I got wrestled down onto the floor by them as they forcefully take away that piece of paper in my hand. These guys...these techniques aren't moves that amateurs can do!

“Alright, this no.6 will be auctioned.”

“VERY GOOD, NO MATTER HOW MUCH YOU OFFER, MIHARU WILL BUY IT!”

“““WHY IN THE WORLD IS SHIMIZU-SAN HERE!?”””

“Sorry, but this is a class affair — hm?”

“No, this..isn't 6. It's 9. Yoshii just looked at it in the opposite way.”

Sugawa-san opens the number in his hand. The number 9 has an arrow to indicate how it should be read.

“What the hell, so it's 9. That shocked us...”

“Yoshii really loves to create commotions.”

“Causing us to spend so much effort on this.”

“It's all Yoshii's fault for being unable to see such a simple number like that. How stupid.”

The FFF members grumbled as they turn away. That was close. If I really got 6, I would have been killed...

“Well, Aki...”

While I'm stroking my joints, Minami walks up to me.

“Hm? Minami, what's up?”

“What should I say? Well...before the meet, why don't we tie our legs toward and practice a bit? We need to practice on chemistry, so I think we should practice to match our breathing and rhythm...”

Minami starts to warm up as she asked me.

Practice for the 2-persons-3-legged race? I see. So Minami's seeing that as really important and actually willing to make a contribution to the class like that. That's nice. In contrast, those classmates of mine who would execute me out of jealousy are really rotten until I want them to the mud in their nails.

“Ah...but I feel that it's best to train with your own partner, right? It's more effective to train together with your partner.”

“I'm not deliberately closing my distance with your body—eh? What did you just say?”

“I just got 9. Didn't you get 6, Minami?”

I open my paper I just drew to Minami for her to see.

That's a pity. If I could partner with the athletic Minami, I can perform better in the 2-persons-3-legged race. And besides, there would be a lot of happy things happening as well...

“Oh, I see. So it's like that...if that's the case, it doesn't matter...”

Hearing me say that, Minami shrugs her shoulders and looks lost as she walks away. Did she feel that it would be easier to pair up with me? We do work together well. If she really thinks this way, it would be really great.

“I'm sorry, Minami-chan. The truth is that, I heaved a sigh of relief just now.”

“Don't worry Mizuki...I think I would do the same thing if our roles were reversed...”

Minami walked to Himeji-san, and both of them said something. It seems to be something involving girls. I shouldn't eavesdrop, and besides, it's time to go back to Yuuji and the rest.

After keeping the number I opened properly, I turn around to return to my usual friends. Yuuji and the rest seem to have finished discussing about the baseball match as Yuuji passed me my batting order number once I got there.

“What's wrong, Akihisa? You didn't get the number you want?”

Yuuji mocked me. If I really got the number I wanted, I would have been buried, right? In a certain sense, maybe such an outcome is really the jackpot.

Speaking of which, who's my partner?

“Oh ya, Yuuji, what's your number?”

“Hm? Well, I haven't drawn yet. I wanted to draw once those guys calm down, but I forgot.”

“Me too.”

“...Same here.”

It seems like these people haven't decided on their partners in the 2-persons-3-legged race. Since the partner number 9 hasn't drawn his number yet, one of these three people will be my partner. If that's the case...

“Please, Hideyoshi! You must draw 9!”

“Hm? What's wrong, Akihisa? You want to be partners with me?”

“Of course! I really want to be partners with Hideyoshi!”

If it's just Yuuji, Hideyoshi and Muttsurini, I would choose Hideyoshi without exception. Considering the honor of the class, it would be better to choose Yuuji or Muttsurini, but things aren't that simple. This world has things that are more important than winning.

“We should draw our numbers too. Besides, the class challenges is more important than the baseball tournament as per schedule.”

“...It will be bad if we don't take part in those events.”

“I feel the same.”

The 3 of them head to the box. As things go, only those 3 haven't drawn numbers yet. The only ones with an undecided partner are me, Himeji-san and Minami. In other words, once they draw the numbers, there will be tragedies of bloodshed.

While everyone's trying to hold back their saliva, Hideyoshi reaches into the box.

“Uu, I got 9--”

“BRING IT ON, YOU GUYS!! I'LL DEFEND THIS PIECE OF PAPER EVEN IF I DIE!!!”

“““DON'T THINK THAT YOU CAN LIVE TO SEE THE SUN TOMORROW, IDIOT!!!”””

Not willing to lose to the overwhelming killing intent, I shouted out and gripped my fists tightly. There's only a few minutes before the 2-persons-3-legged race begins. Until then, I must survive!

“--Sorry, saw it wrongly. It's 6.”

“““...”””

The killing intent immediately dissipated. At this same time, my enthusiasm disappeared. If it's 6, then Hideyoshi's partnered with Minami...I feel that it's a pity, but there's a sense of relief here...

If that's the case, there's only 7 and 9 left. The next one will either be my partner or Himeji-san.

“...(sssst).”

Muttsurini reaches into the box. To him, this is a battle of winning and losing, right? If he can be with Himeji-san, he can touch Himeji-san's amazingly self-aware parts. That guy will jump into the vortex of happiness even when he knows that he would die. I don't want him to get such an envious outcome!

Muttsurini finally pulls out that piece of paper that decides his fate and slowly opens it. Now, what will he get...

“...9--”

“FAREWELL!!”

“DON'T YOU DARE ESCAPE!! EVERYONE, CATCH SAKAMOTO AND OFFER HIM FOR A BLOOD SACRIFICE!!!”

“OSSSUU!!”

Less than a second after it happened, the moment Yuuji knew that Muttsurini's my partner, he realized that he would be Himeji-san's partner without drawing the numbers, and the class immediately ran off in a lightning quick hunting action. Mn, it's not that I don't understand everyone's feelings for wanted to catch that damned guy and execute him.

Looking at Yuuji and the other classmates sprint at a speed that's worthy of a sports meet, Hideyoshi and Muttsurini came up to me with their drawn numbers.

“Really, everyone's so energetic.”

“...That's really noisy.”

“What's wrong, Akihisa? Aren't you going to deal with Yuuji?”

Perhaps suspecting my calmness, Hideyoshi couldn't help but ask. Uu, I want to deal with him on my own...

“No problems. Even if I don't, Kirishima-san will--”

“..I won't forgive you for fooling around.”

“GUUUUAAHHH!!! SHOUKO!? WHERE DID YOU COME OUT FROM!!”

“--Kirishima-san will fulfill my wish for me.”

“I see. Either way, the way I see it, Yuuji most probably doesn't have much of his life left...”

Kirishima-san, who suddenly appeared out of nowhere, has already inserted her slender fingers forcefully into Yuuji's face. That should hurt...

“...Oh yes, Yuuji.”

“Oi, Shouko, hold on a minute! Who would talk casually in this situation? Normally, people would remove the force on the hand, right?”

“...Did auntie give you anything?”

“Hm? You're talking about my mom? Ahh, if that's the case...”

Yuuji, whose face is locked in by the fingers is still talking to Kirishima-san, muttering 'hold on a minute'. But for Yuuji to be able to talk in this situation, I think that's pretty impressive.

“That spot-check, I handed that thing my mom wanted to pass to you in the bag as well.”

“...Bag?”

“Eh, the bag with the hypnotism book.”

“...Really?”

“Yes.”

“...You aren't lying to me?”

“I didn't.”

“...”

“Shouko, what's with you? What will happen if that bag's confiscated?”

“...To me...what...”

“So, what in the world are you saying—”

“...What...did you do to me!

“UWAAAHHH!! IT HURTS, THIS HURRRRTTTTSSSSS!!!”

“...Everything in that bag got confiscated....”

“UWAAAAHHHHH!!!!”

I can vaguely hear a snap. The next moment, I found Yuuji lying limply on the floor.

“...Yuuji is...a big idiot!”

Leaving behind the unconscious Yuuji, Kirishima-san ran off without turning back.

Really, it can't be helped. At this moment, I can only take care of Yuuji. If I'm to keep him lying there, it'll be a huge disturbance to everyone else.

“What did you do, Yuuji?”

I lean over and grab Yuuji's hand to pull him up, asking about the situation just now. Yuuji shakes his head before answering. It doesn't matter anyway, but for Yuuji to be able to get up immediately after such an attack, I really feel that he's not normal in any sense.

“Well...it seems that it's because of me. My mom wanted to hand her something, and those magazines got confiscated as well.”

Seeing her so angry, it must be something important.

“Speaking of what my mom would give to her, don't tell me...a marriage agreement!?”

Ahh, I see. Since Yuuji and Kirishima-san are still underaged, both sides have to agree to the marriage. The marriage agreement that she work so hard to get got confiscated. No wonder she's so angry.

“That's too dangerous...if that's the case, I really have to be grateful to that spot-check guu...!”

Yuuji again tumbled to the ground. Behind him, our classmates are holding tasers as they surround him.

“Take him away.”

“““Yes!”””

They lift up Yuuji's lifeless body and walk to a dark corner of the school compound. Now that Yuuji's gone, he will definitely be executed...but never mind. That guy will only makes things more difficult for me if he continues to live. More importantly, I should focus on the 2-persons-3-legged race.

“Anyway, my partner's Muttsurini. Let's do our best.”

“...Let's do our best.”

We hold our hands together forcefully. In terms of speed, Muttsurini's even faster than Yuuji. In a competition, he should be the best partner.

“Kinoshita's with me, right? Let's do our best as well.”

“Ya, I'd say, let's do our best.”

Minami leans over for some reason. Maybe she practiced with Hideyoshi already. Besides her, Himeji-san looks really worried.

“With Sakamoto-kun...I worried that I would drag him now...”

“Ahh, Himeji-san, I guess you don't have to worry.”

Once Yuuji comes back, his body would be in a state that he can't run normally.

Right now, our groupings are 'me and Muttsurini', 'Minami and Hideyoshi' and 'Himeji-san and corpse'. I can't take part in the mixed group, but at least this may be a good thing.

“What's wrong, Aki? Your partner's not Kinoshita, it's Tsuchiya...but you look rather happy!”

“Eh? Re, really?”

The reason why I felt really happy that my partner's Muttsurini is because I would die if I'm to partner with a girl...but it's really tough to say such things. Anyway, better slip through with a prank.

“I guess. Besides, having Muttsurini as a partner is better than having Yuuji.”

“Heh~ why?”

“Isn't Muttsurini rather cute?”

“““...Eh?”””

Minami and Muttsurini use a really serious expression as they asked me.

“Didn't Muttsurini wear a female outfit at the seaside? I find that he looked rather cute that time—ow!”

Saying till halfway, I got knocked lightly on the back of the head by someone. Who is it!?

“...Sorry, Akihisa.”

Turning around, I see Hideyoshi looking rather unhappy as he stood behind me. Eh? Did Hideyoshi just hit me?

“What's with you, Hideyoshi?”

“Uu...actually, I don't really know, but I couldn't help it...”

“Eh? Like that? Well, it doesn't matter anyway, since it doesn't really hurt...”

Maybe Hideyoshi's hand slipped? I can't imagine Hideyoshi hitting people randomly, so he must have hit my accidentally...but thinking about it carefully, he seemed to have done a wrestling move on me...

"Recently, it seems that Akihisa-kun's interests is spreading. I'll feel really bothered..."

"...Pervert.."

"YOU, YOU TWO ARE MISTAKEN! I'M NOT REALLY INTERESTED IN CROSSDRESSED MUTTSURINI, I'M JUST HAPPY THAT IT'S EASY TO WIN WITH HIM AS MY PARTNER!!"

I continued to shout to Himeji-san and Muttsurini, who're standing far away from me and muttering away. Speaking of which, I DON'T WANT TO BE CALLED A PERVERT BY SOMEONE LIKE MUTTSURINI!!!

"Heh~Aki, you're pretty confident."

It seems that my 'easy to win' words made Minami unhappy as she snapped back. What confidence? I'm not so confident anyway...

"Uu, everyone knows how amazing Muttsurini is, and I'm rather good in spots too."

If we're to compete in a 100m dash, I'll lose to Muttsurini by a bit, but my timing would be rather decent. If Muttsurini and I can get along well, it's not hard to beat the rest.

"Humph, and you really said it...so if that's the case--"

"Hm?"

"How about a race with us?"

"Eh? With you two?"

"That's right. Kinoshita and I. I remember each class is allowed to sent two groups up, right?"

Hearing Minami say this, I seem to remember that this is really the case. Ever year has their own race, and for classes A-F, each class is to send two groups, a total of 12.

Since Minami challenged me, it'll be interesting to challenge them.

"Mn, OK, we'll have a race then."

I pondered for a while before agreeing to it.

"Then, the loser will have to be punished."

"Eh?"

Punishment...if it's Minami, what sort of punishment would she make me do? Just thinking about that alone makes me terrified.

"What's wrong, Aki? Don't you have any confidence of beating our girls group?"

"Shimada, are you mistaken? I'm not a girl."

Woah, Minami actually dared to say that. I don't really have much confidence in studies, but I can't lose in sports!

"Of, of course now! Okay, let's do this!"

Thinking about it, I kept competing with Yuuji with regards to all sorts of things. Compared to that guy's dirty and despicable punishment games, this should be a lot easier. Who's afraid!?

"As for the loser's punishment--"

"Mn, say it, I'll win anyway!"

"If I win--"

"Um hm? If you win--"

"--You'll go out with me."

"Eh? Go out with you...a weekend kind of thing? You're going shopping?"

Minami's thinking of asking me to go out and help her carry things, right? I heard that girls will spend a lot of time on shopping, so Minami would ask me to do this? Maybe I have to carry a lot.

"Uu...well, anyway, it's like that. Shopping, eating, watching movies...a lot."

"That, that's...we can finish it off in a day?"

It feels like the schedule's packed. It seems that the punishment game Minami just said is really punishment.

"...And I didn't just mean 'for one day' when I said that."

Minami muttered to herself in a barely audible voice. Is she planning something else?

"Mn...I can go out with you for one day. But on my side, if I win-let me think, I want a good meal."

"Okay, I'll promise you that."

"So is our bet on?"

"Yeah, if I win--you really have to go out with me!"

Minami sounded really serious. Maybe she would force me to eat something really expensive with her or give me some extremely tough project for me, or maybe she would playfully drag me to the lingerie section to buy stuff...

"The 2-persons-3-legged race is starting soon. All second year students, please gather at the line."

At this moment, the announcement can be heard from the loudspeakers.

Mn, I just don't need to lose this battle! Since it's decided, a man's word is a man's word.

"Okay! Oi, Muttsurini, stop standing far away from me like that, we're starting soon! We have to get a good result!"

"...Akihisa, don't get too close to me."

"I SAID IT WAS A MISUNDERSTANDING!"

I could only try my best to explain to Muttsurini, who's keeping his distance away from me, as we head to the starting point of the 2-persons-3-legged race.

At the starting point where many contestants are bustling noisily. According to the agreement, I line up together with 'Minami and Hideyoshi's group' for the showdown.

BAM!

The clear bang of the guns can be heard, and the front row of contestants started moving. We're in the next group, so it's about time to get prepared mentally.

Trying to relax my joints and muscles, I gently shake my limps and simulate the situation of appearing there later when I step up. When I go up, I have to time my steps and breathing and then slowly speed up.

"..."

Standing beside me, Minami's also waiting for the time to start. She seems to be nervous, looking extremely serious as well.

"You're rather serious, Minami."

"Eh!? Do, do I? I'm okay."

"Haha, why are you so nervous?"

"Since, since when am I nervous?"

"...You, you are, a little..."

"See, I said. It's good to be serious for the glory of the class, but if you don't relax, what's winnable will become unwinnable."

Even from afar, I can see that Minami's body is rather stiff due to nervousness. Why is she so enthusiastic about the 2-persons-3-legged race.

"...But I want to win no matter what..."

Minami muttered that softly, and those words enter my ears seemingly casually.

She's looking at the starting line with a serious look. I don't understand why Minami would be so enthusiastic, but since she's treating this so seriously, as the opponent who agreed to this, I should treat this match seriously. This should be a courtesy.

BAM! The gunshot can be heard. The groups of people in front of us rush off. It'll be us next.

"Let's do this, Muttsurini, we'll be the first one to the finish line!"

"Kinoshita, I'll leave it to you then."

"Mn, I'll do my best too."

"Understood. Miharu would do her best too."

After swallowing saliva, we wait for the starting gun. The time to decide the victors is right now.

BAM--the crisp bang can be heard. Muttsurini and I move forward. Left foot first, then right, then left, I count the beats as we take steps. It feels good.

Then, we should slowly increase our speed and finally end up running forward at the fastest speed. At that point, we almost forget that it's a 2-persons-3-persons race as we run off as if we aren't bound to someone else in the first place. With a partner that can breathe and move in the same movement, I even forgot of the other person's presence.

"THAT GROUP...WHY IS IT 3-PERSONS-4-LEGGED?"

It's only now that Minami realizes that beside Hideyoshi, her other foot is tied together with Shimizu-san's.

"Eh? Wha, WHAT'S GOING ON! HOLD ON, MIHARU! WHAT ARE YOU DOING!?"

"Ahh, onee-sama...even when sticking to you, I can't feel that little chest that's sticking so close to your body. This alluring feeling is making Miharu unable to hold it any longer..."

"STOP TOUCHING ME! MOVE AWAY! IT'S IMPORTANT FOR ME TO WIN THIS!"

"What is onee-sama saying? Miharu's doing this for onee-sama's sake!"

"SINCE WHEN IS IT FOR MY OWN SAKE! IF YOU'RE DOING IT FOR MY OWN SAKE--OI, HO, HOL, HOLD ON! DID YOU JUST REACH FOR MY BACK AND UNDID MY BRA HOOK!?"

"No problems! Even if onee-sama doesn't use a bra to support, there won't be any problems at all!"

"YOU'LL REMEMBER THIS!"

"YES! MIHARU WON'T FORGET THIS FEELING!"

"I DON'T MEAN THAT!"

"What should I do in this situation..."

Recovering ourselves, we reached the end first. I guess our group won.

"Haa, haa, haa...Miharu, you..."

"That's wonderful, onee-sama!"

"What should I say, this two-persons-three-legged(?) race made me feel tired..."

We undo the ropes on our feet, and soon Minami's group came running in to the end.

The 3-persons-4-legged participants actually took second. No matter how I look at it, they're more impressive than us.

"MIHARU! WE LOST BECAUSE OF YOU!!"

"Miharu will go off here. We'll meet later, onee-sama!"

"AH! DAMN IT, MIHARU!!"

After bowing to Minami, Shimizu-san glared at me and shot off like a rabbit.

"...Really, she..."

Though angry, Minami gives a wry smile. That's strange? This match was ruined and yet Minami looked okay.

"Are you alright, Minami?"

"Eh? What do you mean?"

"No, well...if Shimizu-san didn't get involved, you two could have gotten first."

"Well...hm, maybe that would be that case."

"Isn't that right?"

They were able to run a 3-persons-4-legged race at that speed. If it were a normal 2-persons-3-legged race, it'll be hard to tell the outcome. But--

"Thinking through it calmly, this may be the better outcome."

Minami looked like she doesn't care, but wasn't she so adamant on winning at the start?

"Uu? What do you mean...by 'this may be better'?"

"If I use such a despicable mean--even if I feel alright now, I may regret it some day. It may be a little too late, but that's how I feel now."

Despicable means? What did she mean by that? Saying that to me who often hang out with those shameless and evil thugs, is she saying that she would do something even more cruel that I would regret? Thinking about that, my body started to tremble. If that were the case, I might really need to thank Shimizu-san.

"Then our bet is invalid because the match doesn't count."

"Eh? No need, I lose this time, so I'll treat you to a meal."

"No need, that's too awkward. Besides, we haven't had a real duel anyway."

"I insist! Or what is it? You're not willing to let me treat?"

"Uu...since you say it, I'll be thankful for that then."

"Mn, very good!"

After saying that, Minami gave me a bright smile.

At this moment, Hideyoshi, who was silent up till now, said rather hesitantly.

"Hm? What's up, Kinoshita?"

"No, what I want to say is...that 'thing' of yours is still open. Don't you need to wear it back properly?"

"Eh?"

Dok dok, Hideyoshi points at his own chest.

To be honest, I've been mindful of that ever since the beginning...

"Ah...no, that's not it! I didn't forget to wear it back because I felt that there's no difference in leaving the bra hook off!"

Minami covered her chest with both hands and rush to the other corner of the school. I see...so Minami didn't realize it because she didn't feel anything even with no support...

"...(collapse)."

In a corner where nobody noticed, Muttsurini passed out due to massive loss of blood.

### Notes

1. Mathematical Induction is basically based on the idea of proving that the equation can hold, so you're acting as a lawyer here...okay, enough wry comments. When doing mathematical induction, do remember to state that k is a natural number when n=k, since you will be penalized for not doing that (personal experience). Also, natural numbers are positive integers, e.g. 1,2,3,4,5 that can be counted, though there's still debate on whether 0 can be used.
On a side note, the equation 1 is derived through a method called Arithmetic Progression.