Amazing story of how a whole establishment doesn't want to recognize an error that was considered correct for so long.

Anyway, the answer is very difficult to give because 9^9^9 is the representation of the number assuming some operations related to the position of digits, but it is not the number itself.

That's because they know you want the truck back.

Computing at a point for which there is no data requires interpolation based on some number of points for which there is data. However you do whatever interpolation is required, numerical differentiation is still better analytically with central differences than either forward or backward single step differences because the truncation error is smaller in the underlying Taylor series, but none of them are reliable with ill conditioned calculations due to limited finite precision arithmetic and errors in the data. That is why smoothing is generally used whatever the source of the data.

One of many similar stories... gotta get back to writing that book... :D

To conserve space, I left the explicit round trip nature as a mere presumption in this forum. Sorry, I couldn't quote the whole thing word-for-word from my failing memory of that 1959 (or so) classroom question, but round trips were intended. The point was the difference between the teacher’s accepted arithmetic answer and its failure to match the reality of the word problem.

If you want a word problem with impossible quantities (unless you scale the fractional quantities to integers) but an arithmetically sound result, try my favorite IFAQ (InFrequently Asked Question) from an Isaac Asimov story:
If 1½ chickens lay 1½ eggs in 1½ days, how many eggs do 9 chickens lay in 9 days?
Figuring how to set that up the first time is the hard part. The arithmetic involved is very simple.

Actually, I did make a sort of half-trip delivery of cargo once.
My friend/former coworker had stashed all his spare belongings in my New Jersey basement along with furniture and a grand piano elsewhere in the house. He moved to Denver and called asking for help to find a company to move his belongings quickly. All the pro movers were extremely pricy. I told him I’d do the packing and moving if he paid for a one-way Ryder truck rental, my travel expenses, one day of my lost salary, and airfare home. I packed the truck and drove the one-way trip. He got his stuff intact in three days for less than half of the best offer from the pros, who would have taken weeks.

Hallelujah and kudos to you for winning. However,
I do have the caveat that the others' answer (999)
should not be counted against them, as they were
not responsible for not having yet been taught a-
bout exponents.
I was not anywhere near being brilliant like
your daughter in school, especially in math. But
I couldn't stand my 8th grade math teacher. I
would try to explain why my answer was right
(though it probably wasn't), but the only reason
I cared is because I wanted to know where the
mistake was, so that I wouldn't make it again.
And the old b---h would interrupt with, "There are
some things you have to take on faith."
The teacher I had in 5th grade had a science
diagram put on the blackboard, with a "lever" and a "fulcrum". However, the "lever" in this pic-
ture was not functioning as such; there that
lever was on the board, with the triangle (ful-
crum) sitting flat on top of it. When I objected
to this on the ground that it didn't make sense,
she just said that that was the way it was in
the book (not one of ours, I think it was a sort of
teacher's manual), and that was how it was go-
ing to stay. And I am expected to have any
respect for education after that?!--Forget it.

"Is the exponent an operator? We denote it via position and often"
I often use the caret (^) when typing. If it's a a natural anitlog, e^[large expression], on paper I write exp(large expression).

I agree that teaching math as something where you plug and chug and get the one and only answer is wrong. More and more those problems are being solved by computers. In some ways our schools still teach kids to get on the train track that leads from one level of education to the next and then to a stable job. My 7-y/o's teacher told him this narrative. I told him maybe that was true when this teacher was a young woman, but it's less and less true today and it won't be true when you grow up.

Is the exponent an operator? We denote it via position and often, but not always size. But then, even the number 999 (base 10) uses positional arrangement to denote meaning.

One thing this entire discussion shows is that the idea of there being one right answer isn't as solid as we normally believe with math.

"the 'right' answer I was told it was 0 because 'we hadn't been taught negative numbers'."
That's really bad b/c the teacher is just patently wrong.

"But the question didn't have operators, so you can't legitimately do that."
I was considering the exponent as an operator, one that we commonly denote with the superscript. So in my mind if the question does not allow the factorial operator, it does not allow exponents either.

WOW! Reading this story literally brought me to tears! The love of a father for his daughter being ridiculed for being brilliant and the total disregard of the "school system" for a brilliant mind. Cheers to this little girl and her father!

It's a question of definitions. If a trip is defined as "there and back again" (apologies to J.R.R. Tolkien) or a full trip then a 1/2 unit is not a trip. Thus 5 trips would be necessary. Alternatively, one could state the answer as 4 trips (full there and back agains) and a one-way delivery. The other answer would be 9 trips with a trip defined as merely one leg.

I can tell you that none of the people in my trucking department would answer 4-1/2!

Agree with your assertions; however, the example in question was computing the slope at a point for which there is no data. I'd have to go think about it a bit, but I believe a weighted average (depending on the proximity of the points is better than using the point on either side). This type of data is very sparse, and rarely non-monotonic. As Dr. Brenner points out, derivatives are noisy. I seriously doubt with all the other factors not represented in the data (location, time of year, time of day, type of people) that a little averaging will not help better represent the actual information being sought. This is why it is macroeconomics, not physics.

LOL for the earnings forecast!

Jan, I'd like to gently point out that it's not the Math ability she was showing, it was, if anything, the Lateral Thinking that can help bring her success and rewards in whatever field she finds to be the most fun.
I've got an EE degree with about 48 years of rust and dust on it, but 'the way MY mind is wired' (I don't know how else to describe it...) there are a variety of fields in which I'm a whiz at finding solutions!
As y'all have seen here, I love the Socratic Method of inquiry because it can lead me to answers in a wide range of subjects.
There have been lots of books written about "Mavericks" or whatevers who can solve difficult problems that the 'regular professionals can't' and, virtually To A Person, they're shunned by 'regular professionals.'
To the detriment of too many organizations.
So, why ain't I rich? I'm not a good IMPLEMENTER of solutions. Maybe next lifetime...
Cheers, all!

That, of course, reminds me of a work story...
As a relatively new-hired marketing engineer, I was still very shy about 'raising my hand' in situations like this, but well... that's another story, too.

Every month, our division would have a mass meeting where various division managers would summarize the happenings in their departments.

The controller put up a slide that showed the quarter-to-quarter Sales Growth Percentages for the whole division. It was gorgeous! A nearly perfect sawtooth 'curve' varying between nearly identical upper and lower 'limits'.

And then he put up his forecast for the next quarter... the sawtooth was extended into the next quarter with a perfectly horizontal straight line.

I leaned over to my closest friend and whispered, "How much do we pay that asshole to do shit like that?"
He laughed softly and suggested that I not raise that question in public.

I had MANY more opportunities and temptations to ask similar questions over the next 23 or so years with the company, but in the end resigned myself to the fact that there were too few people who had any 'input devices' to assimilate such feedback, let alone understand, 'grok' it or act on it.
Then they offered me a generous retirement package.
As some of you will recall, I snapped it up as fast as I could!

Cheers! and yes, Critical Thinking is Dead.

Half a trip means the truck goes half the distance of a full trip. The problem asked how many trips to deliver the boxes. It didn't say the truck had to return. After 4½ trips all the boxes have been delivered.

In theory, the idea of having a set of things that everyone who goes through the education system knows as a "common core" of information does not seem like a bad idea.

Where it seems to have gone off the rails is in trying to dictate how the information is to be taught instead of saying "x graders should be able to add and subtract", they give these absurd methods for adding and subtracting.

...and I too "specified three digits".

Kansas should know their beef

Interesting story, Thoritsu. ewv's analysis on this one is correct. To further ewv's point, it is pretty rare that the derivative of a data set is smooth enough to reliably differentiate again. Whenever possible, collect data sets as integrals. Congratulations on considering yourself your highest authority.

Voodoo economics uses very sophisticated numerology.

As for your Indian Chief's pronouncement, central differences are 2nd order accuracy (truncation error proportional to the square of the spacing between the tabular points), and both forward and backward differences are first order. But that only works if the function is smooth enough and with arithmetic of sufficient precision. For discrete data samples numerical differentiation is notoriously unstable. Roundoff error alone propagates proportional to the inverse of the spacing size of the data. That is why it is generally best to smooth the data first with something like least squares and then differentiate analytically. Indian Chief should have been thrown out of class and sent back.

it's (9 to the 9th) to the 9th, and here's how that looks:::

196,630,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

-- j
.