Once upon a time, I made reference to 9^9^9 being the largest number that could be expressed using only three digits. Jimmy Hambone Lunddite wrote to inform me that, in fact, 999! was larger than 9^9^9. And basically he's right, because 999! is about 4x102564 while 9^9^9 = 387420489^9 ~= 2x1077.
And this is correct, assuming that 9^9^9 is interpreted as (9^9)^9, which is the correct assumption without additional information. Unfortunately, what I meant to infer was 9^(9^9), which is easier to write if you have nested superscripts (which I could not do in HTML). And 9387420489 is vastly larger than 999!. So let me refine my statement:
9(9^9) is the largest number that can be represented with only three digits.
9^(99!).
Au contraire, I propose one of the Busy Beaver numbers, BB(9^(9^9)). See http://www.scottaaronson.com/writings/bignumbers.html
If you insist on only three non-operator, non-parenthesis characters, then BB(9) is still bigger.
If you insist on nothing but three digits, then I contend that the superscripting metacharacter should count. As in ASCII, where superscripting requires a ^ symbol. And, heck, the factorial ! should count too. Which begs the question of how many bangs I'm allowed to tack on... 99!!!!!!, anyone?
So I'm not sure how we can self-consistently interpret the rules of the game, unless the superscripting metacharacter is considered to be uber-special. But it gave me an excuse to link to Scott's essay, which seemed relevant.
Don't stacked exponents operate from right to left anyway? I think that the extra parentheses in 9^(9^9) aren't necessary, because 9^9^9 = 9^(9^9) != (9^9)^9.
OK, getting back to the original reference of my comment about 9^9^9, I believe what they were trying to say was that this was the largest number that could be represented by writing three decimals and nothing else (positional notation, in this case superscripts, requiring no "ink", if you will).
I think we've determined that if you are allowed to add operators then you can basically factorialize yourself into oblivion.
But Robin does have a point, in that the superscript does imply an operator while not requiring an actual written character. And if that is allowed, then we can simply invent a positional notation that gives us an arbitrarily high number.
So, in conclusion... the problem is ill-posed. Sorry! :)
I may be forced to meme-orialize the phrase "factorialize yourself into oblivion". That said, while the problem may not have been quite well-posed, it's still interesting. Which means that there exists a well-posed problem with an interesting answer, and that problem is recognizably similar to this one.
I think one well-posing of the problem is the one Scott poses in the essay I linked to -- I believe he says, `You're given a pencil, paper, and 15 seconds to write down the biggest well-defined integer that you can. All the mathematical literature can be assumed (i.e., no need to define what "!" or "BB" means), but no self-referentiality ("Twice this number") or undeterminables ("The biggest number you can imagine plus one") or indeterminates ("The mass of Mouser's mom's ass, in picograms").'
This leads fairly directly to uncomputable sequences -- i.e., sequences {X(n)} of numbers such that each one (e.g. X(1) or X(10)) is computable in finite time, but they grow so fast that no algorithm could be written down to compute n -> X(n). That is, you need to write a completely different program for each n.
Which probably isn't what you were thinking of, but is pretty goddamn interesting, IMHO. The fact that uncomputable sequences exist ranks right up there with the existence of a universal code that would allow us to communicate with aliens (vis: algorithmic complexity theory), as far as coolness goes.
Um. My head just blew up! Ouch.
Oh, and can one of you math types tell me how to get the volume on my amplifier to go all the way up to 11?
> Oh, and can one of you math types tell me how to
> get the volume on my amplifier to go all the way up
> to 11?
Aw, that's easy. All you need is some paint and a really little brush. Then you can paint the "11" on right next to the other numbers. Good to go!
if you start the audio control panel on IRIX like this:
apanel --spinaltap
It will have sliders that go to 11. I want sliders that go to 9^(9^9) -that's real power.