#### Comment Preferences

• ##### 2 equations in 3 variables, hmmm(4+ / 0-)
Recommended by:
cfk, Dumbo, Youffraita, Xapulin

It's possible to get down to two variable in one of the equations, but after that, it gets messy to try to prove, at least at a 1st glance.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

[ Parent ]

• ##### If the product(3+ / 0-)

were anything less than 27, say, 26, then there would be multiple solutions.  So basically we have to prove that 27 is a unique maximum with one solution.

• ##### Nine.(3+ / 0-)
Recommended by:
chingchongchinaman, Xapulin, Dumbo

Look, your math is way, WAY beyond my meager skills, but if x=y=z and the answer is 27 then those are 9s.  Or, 3x3.

Over the past 30-odd years, the Democrats have moved to the right, and the Republicans have moved into a mental hospital. --Bill Maher

[ Parent ]

• ##### the catch is that it isn't automatically....(3+ / 0-)
Recommended by:
Youffraita, Xapulin, Dumbo

....assumed that x = y  = z.  You wouldn't need any other equations otherwise.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

[ Parent ]

• ##### It's Dumbo's equation, and he said(2+ / 0-)
Recommended by:
chingchongchinaman, Dumbo

x=y=z=3 is what you have to prove.

3x3x3 -- 33

Like I said, this is way beyond me...but that's how you get 27.

Over the past 30-odd years, the Democrats have moved to the right, and the Republicans have moved into a mental hospital. --Bill Maher

[ Parent ]

• ##### true, but the trick is how(2+ / 0-)
Recommended by:
Youffraita, Dumbo

It's true that x = y = z = 3 is the one solution, but the trick is the proof that this is the only solution.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

[ Parent ]

• ##### See, this is why Dumbo's equation(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

is totally beyond my ken...

You could prove that it's done by multiplying 9 by 3...

but that is still 33

I mean, it's the only solution b/c those are the only things you can multiply and come up with 27.  There are no others.

and when x has to equal y has to equal z and they all have to equal 3...there IS no other answer.

27 is only divisible by three or a multiple of nine.  It's why twenty-six doesn't work...nor does twenty-eight.  We won't even GO to twenty-four.

I'm sure Dumbo can tell me twenty-four times why I haven't provided the proof -- and I'm sure he's right!  I'm just a dumb liberal arts major!

But I'd love to see him explain how to get to twenty-seven without using three squared.

Over the past 30-odd years, the Democrats have moved to the right, and the Republicans have moved into a mental hospital. --Bill Maher

[ Parent ]

• ##### Well, Okay, let me try again.(2+ / 0-)
Recommended by:
Youffraita, chingchongchinaman

I have a box, dimensions or x,y,z.  Or height, length, depth, if you prefer.  Whatever.  When you add up the dimensions, you get 9.  When you multiply them out to get volume, you get 27.

What is the solution?  That's easy to figure out.  Is it the ONLY solution?  Is there some other way you could divvy up the dimensions, like 3.1 x 2.7 x 3.2, for instance, that would give you 27?

The answer is NO, but how you prove it is interesting.  You could use calculus to prove it, which is the obvious way, and that's a real bitch from hell.  So is there an easier non-calc solution?

I'm not sure what THE easiest solution, but mine is this:

Let's start out with the dimensions 3x3x3.  if you increase one of those numbers, you have to make another one smaller.  If I do that with just two numbers, though, it always reduces the volume because of this formula from first year algebra.

(X+k)(X-k) = X^2 - k^2.  Which has to always be less than X^2.  Because you're subtracting out the k^2 shit.

So if I change the dimensions from 3x3 to, oh, say, 3.5x2.5, it should go down right?

3.5x2.5 is the same as (3+0.5)x(3-0.5) which equals 9-0.25.  which is 8.75, the same number we would have got from multiplying out 3.5x2.5 Always less than 9.

SO... Having reminded ourselves of that...  If I have three numbers for the dimensions, and any two of them are unequal, I COULD get a higher volume with same dimensions total by just changing those two numbers to be the average of the two.

And that tells me that the maximum volume can only be achieved if all three dimension numbers, x,y, and z are equal.  Since x+y+z = 9, that means to get the maximum volume, it would have to be x=y=z=9.  Which would give me the maximum volume of x*y*z = 27.

So we know that 27 is the maximum volume, and anything but x=y=z=3 would NOT give us that maximum.  So it HAS to be x=y=z=3, the only solution.

That's my best stab at proving it.  Since it's all done logically, you don't have to resort to advanced calculus.

• ##### Oh, sweetie, yes,(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

YES, I do.  I don't understand one thing you just wrote.  Not one.

The only thing I know is three cubed is twenty-seven, and it's the ONLY answer b/c...it's the only answer.  Twenty-seven is not divisible by very much.  It could be nine divided by three...but that isn't how you set up your equation.

x=y=z=3 is how you set it up.  And to equal twenty-seven, it has to be three cubed.

Am I wrong?

I am pretty dumb at math.  I can add and subtract and multiply...and algebra was always kinda fun...but I never got much beyond that.

Over the past 30-odd years, the Democrats have moved to the right, and the Republicans have moved into a mental hospital. --Bill Maher

[ Parent ]

• ##### Well...(2+ / 0-)
Recommended by:
chingchongchinaman, Youffraita

x+y+z = 8
x*y*z = 18  (instead of 27)

How many solutions are there to that?

Well, there are three easy ones right off the bat.  I could x=y=3, and z=2.  Or x=z=3 and y=2.  or y=z=3 and x=2.

But there could be other solutions as well, couldn't there?  Why do they have to be 3 and 3 and 2?  Might there be some other way to balance them out to get 18?

Because I don't know.  Before tonight, I thought there were probably a lot of other solutions to that, but I'm thinking now there might be something funny about this that straight-jackets you into using whole numbers.

So there might be an easier solution.

• ##### again, the catch is that.....(1+ / 0-)
Recommended by:
Youffraita

.....you've set up only 2 equations with 3 variables, which cannot be simplified sufficiently in of itself to get things down to solve for at least one of the values.  You have to have the same number of equations as variables for some stage of simplification to occur, namely to solve for at least one of the variables by substitution and manipulation of the other equations.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

[ Parent ]

• ##### Dumbo, I'm dumb at higher maths(0+ / 0-)

So maybe I'm straightjacketing myself into that.

I just don't see how x=y=z=3 can ever lead to the solution of 27 without 3 cubed.  If they are all equal, i.e., they all equal three...then you are stuck with my solution.  Which is simplicity itself (but then, when it comes to the maths, as mentioned above: I am simple).

I know I'm missing the finer points of what you and 3CM are saying...but you asked for the simplest solution and that is ALL that I'm capable of.

If you want to go into multiples of two...or fractions thereof...well, I can't play in your sandbox b/c I'm too mathematically stupid.

But even I can do three cubed.  ;-D

Over the past 30-odd years, the Democrats have moved to the right, and the Republicans have moved into a mental hospital. --Bill Maher

[ Parent ]

• ##### 3CM, I am ROfL right now...(2+ / 0-)
Recommended by:
Dumbo, chingchongchinaman

I am NOT good at math, imho, but my boss asked me once & when I said no, he seemed surprised.

B/c I can do paperwork w/out overly relying on the calculator.

Well, and you are roughly my age: we learned to add, subtract, multiply and (sometimes) divide without mechanical help.  So I'm faster than the kids: I  don't need a calculator to add 180 plus 20 and come up with two hundred.

LOL.

Over the past 30-odd years, the Democrats have moved to the right, and the Republicans have moved into a mental hospital. --Bill Maher

[ Parent ]

• ##### true enough about being able to juggle....(0+ / 0-)

.....#'s in one's head for relatively simple, everyday math.

"It's only in books that the officers of the detective force are superior to the weakness of making a mistake." (Wilkie Collins, The Moonstone)

[ Parent ]

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