So fxx*fyy-(fxy)2?
And how is it adapted to a three-dimensional equation? fxxx*fyyy*fzzz-(fxyz)3?
God no. I don't want to type it because it is pure suffering. The determinant of a 3x3 matrix puts the content of the first row as coefficients for determinants of 2x2 matrices constructed from the non-first-row contents of the columns other than the column of the respective coefficient.

This particular test is only one way to optimize functions. When working with a constraint like we are, it's best to look at where your gradient of the function is parallel to the gradient of your constraint using LaGrange multipliers. From there you can create a system of equations with the partials of your function set equal to lambda times the partials of your constraint, as well as your constraint, solve it (by finding relationships b/n x, y, z through lambda), and then you compare the values of your function at your solved-for points.
 
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Plus, if it's just for finding the nature critical points, how would we account for the fact that, since x*y*z has to =1000000, our minimum might not be at what is otherwise a critical point.
If you tested a non-critical point using the determinant of the hessian, you could draw some vague conclusion about your function. I think. The nature of the test is such that it tells you whether the function is dominated by its concavity or by its "twisting." So you could guess whether or not you were closer to a saddle point or to a min/max? I'm not sure. It's not often that you test a non-critical point in the first place. Usually you've looked at where critical points are first.
 
Either you use the equation to eliminate one variable, giving you two free ones, or you use a thingy called a lagrangian multiplier.
The first one sounds much simpler. z=1000000/xy

For the zig-zag net that is a rectangular area of
(3000000/xy+x+y+5)*(x+y+5)
That gets a critical point of (91.6532 , 91.6532 , 119.0432), with a area of 102709.1116 mm2​
Which is a bit better than the simple cube which would have gotten 103525 mm2​

For the cross net that is a rectangular area of
(2000000/xy+2y+5)*(2000000/xy+x+10)
Critical point of (204.726 , 97.7093 , 49.9909), with a area of 94538.4 mm2​
Which is much better than what you get from a perfect cube, 125550 mm2​

Critical points found with the help of Wolfram Alpha.
 
You know, MC's doing this on paper. That must be a real pain.
Yes, but it's do-able. Got to the point where I understood what must be done before I hopped over to Wolfram Alpha. I could probably find those critical points by hand, but I REALLY didn't want to do all the needly math.

Plus, I kinda wanted to know the answer even if the MC doesn't, I'm a math geek currently in Calculus 3 so am learning stuff like this.
 
God no. I don't want to type it because it is pure suffering. The determinant of a 3x3 matrix puts the content of the first row as coefficients for determinants of 2x2 matrices constructed from the non-first-row contents of the columns other than the column of the respective coefficient.

That really wasn't too bad. The matricides are a bit of a mess this way but its readable.

Edit: And it ate my formating... fuck. Nevermind.
 
Hmm, now that you mention it, a net with a zig-zag shape would fold into a cube, with only a area of 2 squares by 5 squares, rather than most other formations that get a 3x4.
XXX
....XXX
Kinda like that, Xs marking the squares. Which gives the equation for the cardboard area being (3z+x+y+5)*(x+y+5) which should allow for a much more compact final product.

It is slightly more compact, if only by virtue of the shape connecting one of the areas where we previously would have needed a flap to seal it together.


With that said, the maths Trondason is doing is far more advanced than I ever learned. I'm forced to stick with basic ninth-grade geometry and algebra, and I was never particularly good even with those. Sorry.

I'll just have to assume his maths is correct.
 
Yes, but it's do-able. Got to the point where I understood what must be done before I hopped over to Wolfram Alpha. I could probably find those critical points by hand, but I REALLY didn't want to do all the needly math.

Plus, I kinda wanted to know the answer even if the MC doesn't, I'm a math geek currently in Calculus 3 so am learning stuff like this.
This actually from a subject called Engineering Maths 1, for anyone curious. It's solvable with advanced high school mathematics, if quite hard for that level, though your method looks like it's working.
 
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The Sleeper Awakens
The Sleeper Awakens

You, in your own unbiased estimation, are a pretty damn fine swordswoman considering you woke up dead with no memories six days ago. That doesn't mean you could take on some asshole who'd been training with a sword since before he could walk in a straight up swordfight. You'll need to get in close to him, too close for a proper swordfight.

He's settling into a stance of his own, and your eyes dart to his feet. He's gonna attack from the left, a thrust to the face. You felt that thought flash into your mind, just like the written test. Thanking whatever supernatural source of insight you had, you waited for the instructor to begin the match. With a shout, he did so.

You rushed forwards, raising your sword and batting away your opponent's thrust. He's too close to stop you from grabbing his wrist, or so you thought only to have him smash his knee into your stomach. You roll backwards, ducking out of the way of his retaliatory swing.

"Your tavern-brawl skills won't help you against a real swordswoman, Takamachi." He said, his face contorted into an ugly sneer. This guy had real skills - the close-combat skills you'd used to beat the two extortionists wouldn't be enough to carry you to victory here. So you looked inwards, towards the source of your unnatural insight, and you lean into the instincts you didn't even know you had. He swings at you again, and you parry it. You thrust forward, and are deflected.

Your eyes meet your opponent's as your blades clash once more. The sneer is gone, replaced only by an intense look of concentration. Then the moment passes, and your eyes refocus on what's important. He attacks again and again, driving you back. The edge of the mat draws near, and he's starting to push you back more and more with each attack. You swing wildly, forcing him back for a mere moment, and jump. You seem to almost glide on the air, sailing over his head, and landing in a roll on the other side of the mat.

You leap back up to your feet, and charge once more. Again, you know where he will strike at you. Again you deflect it. His knee comes sailing for you, and at the last moment you twist out of the way. Your hand grabs behind his knee, and you pull as hard as you can. He goes up into the air and down, but strikes at you with his sword.

You dodge, wind rushing past your ear as his sword goes by. He's back on his feet now, but disoriented for just a moment. You don't let him recover, and swing at his chest. He blocks, and yet another crack of wood smashing against wood rings out. Again you strike, pushing him back. Each time, you attacker harder and harder. His sword begins to chip, splinters flying as he's forced to edge of the mat. You strike, and as he stands his ground your blades clash. Wood against wood, you put all your effort into the blow and with a wordless cry, your strike smashes through his blade and connects with his chest. Amidst a hail of splintered wood, he's sent flying backwards off the mat by the force of your strike.

In a moment of post battle euphoria, you extended your hand to your downed opponent. He grabs it, and you help him up. His face still shows his shock.

"I admit, Takamachi, I misjudged you. For that, you have my apologies." The noble said, before the two of you bowed and left the mat. You walk over to Chise, smiling. She hugs you, violently, and only after some time are you released.

"That was amazing, Kanade!" She says, after letting go of you.

"It wasn't half bad, yeah." You say, blushing slightly. You can tell people are watching you after that fight - most of them lasted less than a minute, as compared to your own's near ten minute running length. "Let's get out of here, yeah?" You ask her, and she nods. The two of you turn and leave the hall, following the directions of the instructors and older students, who were out in force. You ended up in a line in a corridor, for what you weren't sure of.

"Hey, do you know what we're lining up for?" The student in front of you asked.

"No idea." You say, and listen in to the chain of conversation making it's way towards you. After a short while, it becomes clear to you that this is the line to receive your grades, and your roommate (in that it was determined by the order you got your grades in). You turn to Chise, but before you could speak she's started too.

"Would you mind, I mean would you like to room with me?" She asked, blushing and stuttering like she had earlier.

"Sure, Red." You say, smiling reassuringly at her. The two of you while away the hour long wait in line with idle conversation, most of which boils down to Chise telling you what she knows about the Soul Society. You now had a vague idea of what it was, as a whole, though you didn't know any more about what Shinigami actually did, apart from fight nebulously defined monsters known as 'Hollows'.

The line had progressed to the point that you were now able to see the curved desk behind which the instructor handing out grades sat. A small machine seemed to print them on demand, which made you curious. Here, they had such technology despite the primitiveness of Rukongai?

Regardless, it was now your turn to receive your grades. You marched up to the desk, and stated your name. The instructor wordlessly handed it to you, then told you to wait for a moment. Chise was given hers, and then you were assigned a room - number 516-R2. You would later learn that the 'R2' stood for 'Residential Building Two'. You and Chise had qualified for the advanced class, though only you had passed all three tests. The two of you eventually managed to find your way to your room, and after an exhausting first day you opened it's door.

Inside, there were two beds, two closets and two desks. It was the nicest bedroom you - or Chise, for that matter - had ever seen. You flopped down on a bed, and though you had a vague inkling that it wasn't what some of your classmates might be used too, it still felt positively palatial.

You have classes starting tomorrow, but what other actions will you undertake during the first week? Choose two:

[x] Socialise:
[x] With Chise [Unlocks it as a free action]
[x] With students from Rukongai
[x] With students in general​
[x] Investigate your supernatural insight
[x] Train on your own time:
[x] Hakuda
[x] Hoho Theory
[x] Kido Theory
[x] Zanjutsu
[x] Reiryoku Control
A/N: Character sheet should be up soon. Also, yes @Trondason did solve the question.
 
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[X] Socialise:
- [X] With Chise [Unlocks it as a free action]
[X] Train on your own time:
- [X] Reiryoku Control
-- [X] Find out your power level

I would suggest Kido Theory - Hado type, or Zanjutsu, but first we need to find out just how much power we're dealing with to start.
 
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[X] Socialise:
- [X] With Chise [Unlocks it as a free action]
[X] Train on your own time:
- [X] Reiryoku Control
-- [X] Find out your power level
 
Character Sheet
Character Sheet:


Name: Kanade Takamachi
Gender: Female
Physical Age: 20

Stats

Reiryoku: 43 {+10}
Strength: 14 [52]
Endurance: 10 [40]
Speed: 12 [48]
Intelligence: 15
Knowledge: ?? [15]*

*Due to the 'Unknown Insight' trait, the true value of this stat has been hidden. Numbers in {} are the result of Reiryoku Gathering.

Skills:

Hakuda: 20
- Brawling: 10
Hoho: 10
- Sonido: 1
Kido: 10
Zanjustu: 25
Reiryoku Gathering: 10
Reiryoku Channelling: 10
Reiryoku Control: 10
Shinigami Duties Knowledge: 9
Socialise: 12
Spiritual Awareness: 9

Traits:

Amnesia: You don't remember anything about your life as a living human. This is extremely unusual.

Supernatural Insight: You know things, have instincts, you shouldn't.

High Speed Regeneration: You may regenerate from almost any wound, provided it does not destroy your brain and that you have enough reiryoku. Currently, at full power you could regenerate a hand.

End-State: ???

Notable Items:

Asauchi: You have started to bond with what will one day be your zanpakuto. 750/????
 
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[X] Socialise:
- [X] With Chise [Unlocks it as a free action]
[X] Train on your own time:
- [X] Reiryoku Control
-- [X] Find out your power level
 
[X] Socialise:
-[X] With Chise [Unlocks it as a free action]
[X] Investigate your supernatural insight
 
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