I am reminded of my own experiences in math edjimukashun.

In grade school, we had this problem: There is a set of identical boxes to move by truck. The truck can hold 4 boxes at a time. How many trips does it take to move 18 boxes. I had quite an argument with the teacher, who insisted that the answer was 4½. I said that I understood simple division and fractions, but the right answer is 5. The truck requires a full trip to move the last 2 boxes. There is no such thing as half a trip. But she wouldn't admit that it took 5 trips.

Then in high school algebra, we were asked for the factors of a²-b². They are (a+b) (a-b). Then the teacher asked if you can factor a²+b². The rest of the class said no, and I said yes. The teacher turned to me and said, "You can't factor that," and I told her, "Sure you can. (a+ib) and (a-ib)." She gave me one of her angled glances and said, "We're not using complex numbers in this class."
It wasn't a battle. We had a mutual understanding that my answer was right but I was being a math wise-ass, going beyond the scope of what most of the rest of the class could deal with.

If decimal digits are implied (which I think is the case), then the 9s win. If Hexadecimal is accepted, the F^F^F would be larger, but that leads to the next idea, representing bases larger than 16. Do we stop at base 36 that would exhaust the standard English alphabet for single digits.

Nonsense. The problem specified three digits, NOT 3 digits combined with some number of arbitrary written operators, such as the factorial function. Admittedly, factorials increase faster than exponentials, but that wasn't the question. The exponentiation character used in the examples is only an artifice for displaying the positional notation of superscripts, which this forum does not permit as a valid format.
As a curiosity, I'm testing whether the character set here accepts the extended-ASCII square and cube characters (0178 and 0179), as in 4² or 5³. Well, yes, it does.
As to allowing arbitrary operators, I create the arbitrary (trivial) function [Goofus-malarkus(999)] which multiplies 999 by Aleph-sub2, which is the second transfinite cardinality. Multiplying any positive integer by that quantity produces the identical number-of-curves-in-space infinity.

Arguably, you can get an even bigger number than 999 by simply not limiting yourself to the decimal system.

https://www.mathsisfun.com/hexadecima...

And if you use, something like, base 9E9999, then you could get a universe stomping number with just three 9E999adecimal digits. :)

1.9663 times 10 to the 77th

that is 1.9682 times 10 to the 74th (as a
multiplier) larger than 999 -- j
.

this is priceless;;; Thank You Sir! -- j
.

Yes, you are right on the 9^(9^9).
Still smaller than higher base ops, but greater than 9^99.
As you said, in hex the higher digits are A-F. Still digits since they represent numbers imo.

Fun brain food while I'm having morning coffee.
Assuming you're limited to 3 numerical digits but not mathematical operators/notation, I started with the daughter's triple exponential 9^9^9 and after a bit of googling resulted in ~ 2.95e94. But that is obviously less than 9e99. Then I found a factorial calculator which says that 9!9!9 is way bigger than 9e99. Anyone know of any other obscure operators that arrive at a larger answer?

That depends on the meaning of 'digit'. When representing a hexadecimal number we use both letters and digits. Admittedly that's a fuzzy definition but if you define 'digits' as 0-9, then the largest hexadecimal value you can make with three digits is 999.

I may be biased in this because every time I write a hexadecimal parsing routine I treat digits and letters separately.

And isn't 9^9^9 evaluated as 9^(9^9)? The author deals with 9^99 in the comments.

Kansas State 1978 my first semester

Larger base would use a different symbol than 9 for the highest single digit number, so 999 is not correct. The larger base is the largest answer I have heard so far, although choosing a larger base is always possible. So is this actually an undefinable answer?
Also 9^ 99 is greater than 9^9^9, so the answer in the tale should have been shown incorrect by the instructors if they actually cared and thought about it.

What institution of higher education was that, and what year if you don't mind, dbh?

It reminds me when I was asked in Freshman (college) English to draft a menu for an evening. I had chateaubriand as the main course. The teacher (TA) laughed that one person (me) had a wine as the main course. When I pointed out that it was beef, she refused to apologize.

Before reading the article, I came up with the same answer his daughter did. I’m sure many others outside the education system did also. The fact that no other student or teacher in the entire country came up with this answer (or if they did, they kept quiet) is sufficient evidence that our education system is in need of a complete overhaul. A good start would be 100% privatization.

Before I dove in here, I went, well "999… oh, well, 99^9 ah! 9^9^9…" You just have to stop and think. See Thinking Fast and Slow by Daniel Kahneman. Even people who work with statistics professionally show poor intuition with statistical problems from everyday life.

One time in a physics class, we were going over the homework in the usual way: "How do you do Number 5?... What is the answer for Number 7, I got close but not the same one in the back?… What formula do you use for Number 4?" Then the prof stopped. He said, "You kids go out in the back yard and shoot hoops for 45 minutes and not make a single shot and still claim that you had a good time. How long did you spend on Number 5 before you gave up? How many methods did you try??"

I ended up teaching 'the new math' to my 8th grade class because the doddering old nun we had as a teacher was totally adrift - and Wm can affirm how bad I am at math! If the 'bar' is really low (old nun and disinterested students), sometimes it is easy to excel.

I note that the daughter is now majoring in Bio/BioMed. This puzzles me, because she obviously had a talent for math...and Bio is where you head if you love science and are not good at math (did that myself). I suspect that 'the system got to her' eventually.

I thought it was superb that the father took the matter to national level and insisted that all the other answers be marked wrong because many of those students and their parents will now be upset at Common Core!

Jan

Bravo.

Jan

The daughter missed 9! to 9! to 9!.

By the way, I showed this story to one of my employees who had probably been much like that little girl when she was growing up and her answer was: What base?

The digit sequence 999 can be made arbitrarily large by selecting large bases. So, maybe 999 is the right answer after all!

A(9, A(9,9)) Or A(A(9,9), 9) or some such arrangement of Ackermann's function; whichever one goes up faster! Not certain if that's bigger than Graham's number, but it's gotta be a lot closer than 9^9^9 or pitiful little 999.

Some might argue those are functions, but exponentiation is a function too. or at least it's a shorthand representation, but when I think about it so is decimal place notation. 9100+910+9 they're all higher functions, so my 1st answer stands.

This type of thinking has been going on in the education field for as long as I've been around. I went through the same kind of thing when I had to defend my son over an essay he turned in, in the form of an 8mm movie. Educators seem to hate innovation.

But the question didn't have operators, so you can't legitimately do that. (BTW have you done the numbers between 1 and 100 with exactly 4 4s and operators?)

Wikipedia says that you evaluate the exponents top down, which is what he did. Unfortunately that line has a "citation needed" annotation so make of it what you wish.

I agree that requiring everyone to have their answer marked wrong was a bit rude, although technically they were. The inherent problem is the "one right answer" which is clearly a rather important part of math but the application gets fuzzy when testing someone to determine if they understand what's been taught.

I had an encounter with a grade school teacher who asked what was 1 minus 2. I confidently wrote down -1 and was told I was wrong. At my incredulous request for the 'right' answer I was told it was 0 because "we hadn't been taught negative numbers". I didn't buy that argument then and I don't buy it now. Math is math whether you've been taught it or not!

Enjoyed that story...thanks

"how he battled the entire educational system over his daughter's answer."
I get what he's saying, but IMHO he's fighting the wrong battle and being annoying little $hit. I admire his daughter for thinking "outside-the-box". If we open the question to operators, what about using the factorial operator? In the article he appears to do the operation on the right first? Is this correct precedence? Thinking of creative ways to get larger numbers using operators is just the kind of nerd discussion I'd be interested in.

Making it a state case to get her score changed comes of as smart a$$. Then demanding they mark all other answers incorrect (b/c you said, only one answer can be correct. you said it! you said it!) is venturing into dumb a$$. So I find the actual discussion interesting, but the pissing contest with the school really stupid. Just focus on the math, science, and life, and ignore the minutia of testing scoring rules.

So foolish. Although I see value in standardized tests, wrong is wrong. Just admit it and move on. One of my favorite teachers, Dr Armand Dilpare at FIT, told me that there are three stages of learning:
1. Learning a thing
2. Doing a thing
3. Teaching a thing to others
...and you learn more about the thing in each stage. Teachers that know it all are both poor teachers and are missing out.

The story also reminds me of an incident as an undergraduate senior in macroeconomics as an elective. We were talking about elasticity of supply and demand by calculating the slope of the supply/demand curve. The graduate student teacher asserted that a forward difference was more accurate than a backwards or central difference for estimating the slope. After the class, I went to see her and noted that depending on the data, a central difference may likely be more accurate. She became hostile, and after raising her voice and reasserting her brilliance, she said, "In my country (India) students do not question the word of teachers". By that time I was irritated too, and reminded her she is not in her country, and in this country one has to be correct to be correct. One does not get to be right because on one's position. She promptly threw me out of her class, and told me not to come back. I had to go see the head of the business school department (the tail on the engineering dog at Dr Brenner's FIT back in the day) to get back in. He agreed to proctor the final and give me that grade for the class (3 of my last 6 credits). I got an A, and that was the last argument I had with a business major about math. However, one of my buddies with a PhD in EE, went and got his MBA at Chicago. He said the correlation mathematics he did there were harder than anything he did in graduate school.