## 0759: "3x9"

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kg333
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### 0759: "3x9"

Alt text: "Handy exam trick: when you know the answer but not the correct derivation, derive blindly forward from the givens and backward from the answer, and join the chains once the equations start looking similar. Sometimes the graders don't notice the seam."

I don't think I've ever seen that type of derivation before...

intertubes
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### Re: "3x9" Discussion (#759)

Wait, why is he dividing into a square root symbol? /doesn't get that

Although when it comes to the alt text, I've done such a thing on a test before. It works!
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masamune55
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### Re: "3x9" Discussion (#759)

*MIND BLOWN*

thegamefreak0134
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### Re: "3x9" Discussion (#759)

And nerdsnipe in 3... 2...

Yes, I may or may not have checked his work. So shoot me.

alemfi
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### Re: "3x9" Discussion (#759)

intertubes wrote: Wait, why is he dividing into a square root symbol? /doesn't get that

It's simply making use of the similarity in appearance of the square root symbol to the way one does long division. With the alt text, it's simply implying such an absurdity might be overlooked by an examiner.

rwald
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### Re: "3x9" Discussion (#759)

Initially this looks like another example of this, but it actually makes sense why this works; any time you multiply x by x^2, you'll get the same result as if you divide x^4 by x.

Omegaton
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### Re: "3x9" Discussion (#759)

Man, I can't believe when I first read this I didn't notice the square root switching to a division symbol was out of the ordinary.

Grumbleduke
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### Re: "3x9" Discussion (#759)

A quick bit of algebra shows that this will work for any two numbers where:
$\begin{eqnarray*}a \times b & = & a \times \sqrt{b^2} \\ & = & a \sqrt{b^2}\\ & = & \frac{b^2}{a} \\ \therefore a^2 & = & b \end{eqnarray*}$

It's not all that exciting; try it with 2 and 4, or -3 and 9...
However, the alt text is very good advice.

Edit: Ah, assuming that a isn't 0 of course - although I think it still works algebraicly, if you use l'Hopital's rule on the fraction.
Last edited by Grumbleduke on Mon Jun 28, 2010 4:04 am UTC, edited 1 time in total.

rog
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### Re: "3x9" Discussion (#759)

Oh good, I came in here just to make sure my brain was supposed to be broken by that.

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CorruptUser
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### Re: "3x9" Discussion (#759)

Must try this on a math exam.

Feak
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### Re: "3x9" Discussion (#759)

Actually, the strategy in the alt text is exactly how I got through physical chemistry

ARandomDude
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### Re: "3x9" Discussion (#759)

I was going to say that this works for all a * a^2, except I gotten beaten to it. I feel dumb again >_>.

Scarlet
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### Re: "3x9" Discussion (#759)

Best alt-text ever. So very true; TA's are lazy after all
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Turtle_
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### Re: "3x9" Discussion (#759)

It's like cancelling the sixes in 16/64. Or the nines in 19/95 or any other similar things. The alt text is a good idea too, I think I've done it before.
"Sometimes lies were more dependable than the truth." ~ Ender's Game
"Ignorance more frequently begets confidence than does knowledge." ~ Charles Darwin

iGeek
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### Re: "3x9" Discussion (#759)

On a physics exam, a friend of mine couldn't remember whether a particular equation had "squared" or "times two" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.

2^2 = 2*2

llyralei
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### Re: "3x9" Discussion (#759)

My geometry teacher in high school used to yell at me aaaall the time for taking too many complicated steps for simple problems... She was like a raging bull when it came to grading tests; I still have nightmares of her red marker yelling at me from the test papers.

Turtle_
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### Re: "3x9" Discussion (#759)

iGeek wrote:On a physics exam, a friend of mine couldn't remember whether a particular equation had "squared" or "times two" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.

2^2 = 2*2

I did something similar one time. I was trying to remember whether the full law of sines said that sinA / a = R or a / sinA = R, so I tried an example and got sinA / a seemed like R. The real answer was a / sinA = 2R
"Sometimes lies were more dependable than the truth." ~ Ender's Game
"Ignorance more frequently begets confidence than does knowledge." ~ Charles Darwin

djagir
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### Re: "3x9" Discussion (#759)

I've had the alt text work for me, too. The magic phrase at the seam was, "Given (previous result), it can be shown that..."

ChH
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### Re: "3x9" Discussion (#759)

Oh ... that's awesome.
I remember a "physics for electrical engineering students" test where we were asked to derive the capacitance of a coaxial cable. I had the basic formula for capacitance and the formula for capacitance of a coax cable (the answer) in my one page of notes we were allowed to bring in, and did exactly what he showed - worked down half-way from the start, and up the rest of the way from the answer, and there was a serious "seam" where they met. Got full credit, though. And to date myself ... that was in 1991 or 1992 ...

ringobob
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### Re: "3x9" Discussion (#759)

Took me a second (or 30) to figure out what I was looking at... my favorite exam answer was one on a diff-eq test word problem with James Bond wearing a jet pack that ran out of gas, I was supposed to figure out his speed at a certain point... I didn't reach a good point to put a seam, but I did get partial credit for pointing out that my work showed him getting sent into the stratosphere quickly after running out of gas (luckily the prof had a sense of humor). I looked everywhere for the sign flip, but couldn't find it in time.

Initially this looks like another example of this, but it actually makes sense why this works
I'm not convinced that that one doesn't have a more satisfying explanation. Took me forever to come up with a human satisfying explanation for the commutative property of multiplication.

REDace0
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### Re: "3x9" Discussion (#759)

As a math grader, I would probably let things like that slide.

Definitely not that in particular, but similar things on ODE exams I'm sure I wouldn't notice.

Shay Guy
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### Re: "3x9" Discussion (#759)

iGeek wrote:On a physics exam, a friend of mine couldn't remember whether a particular equation had "squared" or "times two" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.

2^2 = 2*2

You've heard of Graham's number, right? That absurdly huge number equivalent to 3↑↑↑...↑↑↑3, with a much smaller but still absurdly huge number of up arrows?

Change those 3s to 2s, and it reduces to 4.

Two and two is four, no matter how you do it.

ringobob wrote:I'm not convinced that that one doesn't have a more satisfying explanation. Took me forever to come up with a human satisfying explanation for the commutative property of multiplication.

What's so hard about that? Just think of a rectangle divided into either its rows or its columns.

SorryBoringNickName
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### Re: "3x9" Discussion (#759)

uh...
ok...

....ah...

what?....

i'm sorry....

huh?

I think I need to go to the park now.

Or something.
Nothing to do, nowhere to go-oh

Arancaytar
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### Re: "3x9" Discussion (#759)

In order to multiply by three, he is squaring and then dividing by three. The only reason this happens to work is that 9 is the square of three.

Also, "show your work"-type problems are stupid.

$\frac{49}{98} = \frac{4\not{9}}{\not{9}8} = \frac{4}{8} = \frac{1}{2}$

"You cannot dual-wield the sharks. One is enough." -Our DM.

ringobob
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### Re: "3x9" Discussion (#759)

What's so hard about that? Just think of a rectangle divided into either its rows or its columns.
Nothing's hard about that. It really only took me probably 45 minutes... in 2-3 minute chunks spread out over 5+ years. I always just accepted it, wondered on it briefly every now and then, and then finally visualized it similarly to what you said.

Nereid
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### Re: "3x9" Discussion (#759)

Turtle_ wrote:I did something similar one time. I was trying to remember whether the full law of sines said that sinA / a = R or a / sinA = R, so I tried an example and got sinA / a seemed like R. The real answer was a / sinA = 2R

Dimensional analysis is your friend.

Turtle_
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### Re: "3x9" Discussion (#759)

Nereid wrote:
Turtle_ wrote:I did something similar one time. I was trying to remember whether the full law of sines said that sinA / a = R or a / sinA = R, so I tried an example and got sinA / a seemed like R. The real answer was a / sinA = 2R
Dimensional analysis is your friend.

I still would have missed the most important "2R" part.

Shay Guy wrote:
iGeek wrote:On a physics exam, a friend of mine couldn't remember whether a particular equation had "squared" or "times two" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.

2^2 = 2*2

You've heard of Graham's number, right? That absurdly huge number equivalent to 3↑↑↑...↑↑↑3, with a much smaller but still absurdly huge number of up arrows?
Change those 3s to 2s, and it reduces to 4.
Two and two is four, no matter how you do it.

Graham's number is actually really funny when you learn a bit about it. It's the upper bound to a problem. The original lower bound was six. Thanks to the wonders of modern math however, we have raised it to eleven. It's basically "well we know the number is between 11 and a number so huge it's not much different from having no upper bound at all."
"Sometimes lies were more dependable than the truth." ~ Ender's Game
"Ignorance more frequently begets confidence than does knowledge." ~ Charles Darwin

asdfzxc
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### Re: "3x9" Discussion (#759)

Holy crap, a math comic that I can actually make sense of!

happysteve
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### Re: "3x9" Discussion (#759)

Kinda reminds me of this fun math tidbit...

Update: Aww crud, I messed that up. Sorry, it was late when I wrote that. Dangit, how the heck did that go again... grrr.... oh yeah.

[imath]30 - x - x^2 = 10[/imath]
[imath](6+x)(5-x) = 10[/imath]
[imath]x = [4, -5][/imath]

There. *sigh* I need more sleep.
Last edited by happysteve on Mon Jun 28, 2010 5:50 pm UTC, edited 2 times in total.

Shivari
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### Re: "3x9" Discussion (#759)

happysteve wrote:Kinda reminds me of this fun math tidbit...

Solve for x:
[imath]x^2 - x - 20 = 10[/imath] ... hmm okay, factor the left hand side...
[imath](x + 4)(x - 5) = 10[/imath]
ah great, now it's just a matter of solving for (x + 4) = 10 and (x - 5) = 10
x is either 6 or -5
Check with the original statement:
[imath]6^2 - 6 - 20 = 10[/imath] ... yup, that works
[imath](-5)^2 - (-5) - 20 = 10[/imath] ... yup, that works too.

yay, problem solved.

... I missed that part where that was interesting and not just what you always do in that scenario.

rjk1994
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### Re: "3x9" Discussion (#759)

Shay Guy wrote:
You've heard of Graham's number, right? That absurdly huge number equivalent to 3↑↑↑...↑↑↑3, with a much smaller but still absurdly huge number of up arrows?

Change those 3s to 2s, and it reduces to 4.

Two and two is four, no matter how you do it.

What? Are you serious? I really don't think it does.

2^2 = 4
2^2^2 = 2^4 = 16
2^2^2^2 = 2^2^4 = 2^16 = 65536
65536 != 4
Q.E.D.

Anyway, great comic. Wish I'd read this before Thursday's fail of an FP2 exam.

Enormatron
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### Re: "3x9" Discussion (#759)

Grumbleduke wrote:A quick bit of algebra shows that this will work for any two numbers where:
$\begin{eqnarray*}a \times b & = & a \times \sqrt{b^2} \\ & = & a \sqrt{b^2}\\ & = & \frac{b^2}{a} \\ \therefore a^2 & = & b \end{eqnarray*}$

It's not all that exciting; try it with 2 and 4, or -3 and 9...
However, the alt text is very good advice.

Edit: Ah, assuming that a isn't 0 of course - although I think it still works algebraicly, if you use l'Hopital's rule on the fraction.

The algebra initially does work, sort of, but only if you assume this equality
$\begin{eqnarray*}& a \times \sqrt{b^2} & = & \frac{b^2}{a} \end{eqnarray*}$
instead of deriving it.
However, if a=0, then you can't use L'Hopital's Rule. L'Hopital's rule doesn't just let you divide stuff by 0 all willy nilly, it only works when you have a fraction and the numerator and denominator are both nonzero variables, but both are approaching 0. In this case, the fraction you described has numbers in the top and bottom, not variables, and if the numbers are both 0, L'Hopital isn't going to help you.
Last edited by Enormatron on Mon Jun 28, 2010 6:02 am UTC, edited 2 times in total.

skeptical scientist
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### Re: "3x9" Discussion (#759)

xkcd wrote:Alt-text:
Handy exam trick: when you know the answer but not the correct derivation, derive blindly forward from the givens and backward from the answer, and join the chains once the equations start looking similar. Sometimes the graders don't notice the seam.

Damn it Randall! This is why we can't have nice things.

Shivari wrote:... I missed that part where that was interesting and not just what you always do in that scenario.

That's not what you always do in that scenario. It's another example of a really stupid way of solving a problem which produces meaningless gibberish in general but happens to produce the right answer here because of carefully chosen numbers (like canceling the sixes in 16/64). Except here he's cheating because (x-5)=10 has the solution 15, not -5.
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### Re: "3x9" Discussion (#759)

RT
masamune55 wrote:*MIND BLOWN*
I wonder in how many cases this works.
hmm, we have x * y = y*y/x. Which is x*x = y. Makes sense, and it means he got lucky in this case.
If x*x <> y, I wonder what he would have had to do to derive it?
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TripleElation
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### Re: "3x9" Discussion (#759)

Shivari wrote:
happysteve wrote:Kinda reminds me of this fun math tidbit...

Solve for x:
[imath]x^2 - x - 20 = 10[/imath] ... hmm okay, factor the left hand side...
[imath](x + 4)(x - 5) = 10[/imath]
ah great, now it's just a matter of solving for (x + 4) = 10 and (x - 5) = 10
x is either 6 or -5
Check with the original statement:
[imath]6^2 - 6 - 20 = 10[/imath] ... yup, that works
[imath](-5)^2 - (-5) - 20 = 10[/imath] ... yup, that works too.

yay, problem solved.

... I missed that part where that was interesting and not just what you always do in that scenario.

When you get to xy=0 you can conclude that either x=0 or y=0 (given that you're working in an integral domain).

The same is not true of xy=10. For example, 2*5=10 but neither 2=10 nor 5=10

Max2009
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### Re: "3x9" Discussion (#759)

Being a college student, I would like to point out that the alt text method works nearly all the time.
I can only remember one case where the grader wrote "this is the correct answer but I have no idea how you got to it".
And yes, I do that all the time.
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Dvil
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### Re: "3x9" Discussion (#759)

happysteve wrote:Kinda reminds me of this fun math tidbit...

Solve for x:
[imath]x^2 - x - 20 = 10[/imath] ... hmm okay, factor the left hand side...
[imath](x + 4)(x - 5) = 10[/imath]
ah great, now it's just a matter of solving for (x + 4) = 10 and (x - 5) = 10
x is either 6 or -5
Check with the original statement:
[imath]6^2 - 6 - 20 = 10[/imath] ... yup, that works
[imath](-5)^2 - (-5) - 20 = 10[/imath] ... yup, that works too.

yay, problem solved.

While I do see the deliberate mistake, you also seem to be saying that if (x - 5) = 10, then x = -5, where it should really be 15.

Glenn Magus Harvey
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### Re: "3x9" Discussion (#759)

This strip is extremely, extremely epic win.

Best one in a long time.

airshowfan
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### Re: "3x9" Discussion (#759)

I, too, use this "handy trick" ALL THE TIME. "Prove that blah is equivalent to blah" pretty much means "Start working forwards from this one and backwards from that one and see if you can make them meet in the middle".

Well, I do usually manage to close up the seam. And when I don't, the gaping hole is pretty obvious (and I have to resist the temptation to write "Then a miracle happens" into the gap). But I turn the work in one way or the other, so that the grader can see that even when there is a gaping hole, I can at least get some of the steps.

So I guess it's less of a "handy trick", and more of a "technique that usually works but sometimes doesn't work completely". I wonder if I ever pulled off making it LOOK like it worked when in fact it didn't. Probably not. Stupid honesty.

Max2009
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### Re: "3x9" Discussion (#759)

airshowfan wrote:Well, I do usually manage to close up the seam. And when I don't, the gaping hole is pretty obvious (and I have to resist the temptation to write "Then a miracle happens" into the gap)

I did that once.
I actually wrote "and then we conduct a voodoo ritual and transform x into y (or whatever step I was missing).
I got 85% for that one and a big smiley. Proving that even though all mathematicians are crazy, some of them have a sense of humor.
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