I have given you all an index card. You have one minute to write down the biggest number that you can (barring infinity) using any mathematical function that you know (addition, subtraction, exponentation, ackermann, Knuth's up arrow, etc). Go.
This is the answer. I win. It grows so explosively that we aren't sure of anything after BB(4). BB(5), BB(6), and BB(12) listed below are calculated MINIMUMS for the actual answer. The true result is likely far larger.
BB(5) = at least 4098
BB(6) = at least 4.640 × 10^1439
BB(12) = at least 4098 ⇈ 166 (which is 4098 raised to the power of 4098 raised to the power of 4098, etc, in a stack 166 4098's high)
(4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098^4098)
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Anonymous2009-06-26 5:12
Oh, and "BB(4)" is an abbreviation for "Σ(4, 2)". Didn't see that in the wiki article. That's paired with the max shifts function, which is even bigger by definition than the busy beaver function. S(4, 2) is 107, for example.
The BB() format is more compact, though, so you'd be able to write a bigger number more quickly/in less space.
For example, with 10 characters, BB(999999) would most likely be bigger than S(9999,999) and you'd really be shooting in the dark as to the best way to optimize the matrix. So just do BB() with as many 9's as you can put in there.
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Anonymous2009-06-26 5:40
>>6
One more post before I go to bed because the point might not have sunk in for some people.
Few desktop calculators can even handle 4098^4098 (4098 ⇈ 2), let alone 4098^4098^4098 (4098 ⇈ 3).
4098^4098 is a 14,805 digit number. It's too big to fit in a single post.
Consumer-grade hardware isn't even physically capable of calculating 4098 ⇈ 166.
the largest number of relevance to anything in the universe?
I thought that was gogoplex or something.
I believe that number was designed as an estimate as the largest number ever needed to quantify the total number of elements of a set of the single most numerous 'thing' in the universe.
what they decided that 'thing' is, i don't know.
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Anonymous2009-06-27 16:33
>>15
wtf? that's total nonsense. the total number of atoms in the universe is only about 10^120 or so. A googolplex is far larger than that.
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Anonymous2009-06-27 17:12
>>16
>atoms
not the single most numerous thing in the universe.
sorry, perhaps I did not explain this right.
I meant it was a number made ridiculous enough that it would be more than almost anything of relevance to our universe.
I suppose even totaling the number of elements of the single most thing of the universe would even be the largest number of possible relevance.
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Anonymous2009-06-27 23:58
>I suppose even totaling the number of elements of the single most thing of the universe would even be the largest number of possible relevance.
I meant to say:
>I suppose NOT even totaling the number of elements of the single most thing of the universe would be the largest number of possible relevance.
>>18 also, how i BB quote?
Make a space after ">".
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4tran2009-06-28 6:07
>>11 >>13
If you're time (1 minute) or space limited, you're better off stacking on one more BB(...) instead of writing out "Graham's number"
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Anonymous2009-06-28 12:58
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Anonymous2009-06-28 15:34
9000
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Anonymous2009-06-28 16:27
>>21
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>>1
This is sortof a neat experiment, until you try to find out who wins. How do you intend to compare these numbers? Who's going to figure out which is biggest?
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Anonymous2009-06-28 20:44
This famous math activity usually limits the time period to 10-20 seconds, but longer periods can be fun, too. However, OP forgot a rule: Your answer must define a specific integer in such a way that an educated mathematician can, given infinite time, reliably arrive at your integer with only the information on your card. That rules out answers like "infinity", "Jack's answer + 1", or "the biggest number ever imagined + 1".
Graham's Number
In order to use Graham's Number, you would have to define it, because it has no universal variable. If you have to define it, you might as well use something better. See below.
BB(BB(...BB(999)...))
While nested Busy Beavers seems a natural extension, no one has yet (to my knowledge/googling) published a serious peer-reviewed paper doing so. Meaning you've got nothing to cite if someone wants to challenge you on it.
I think, given a very tight time limit, the best solution would be something like the following, getting in as many 1's as possible:
BB(1111111111111111111111)
This is bigger than BB(1111) ↑↑↑↑ BB(1111) but took the same amount of time to write. 1's are faster to write than 9's and ↑'s took me about twice as long to write as 1's. If you've got a minute or longer, I would suggest defining a Graham-esque number utilizing Busy Beavers. Something like the following, perhaps:
Answer = aBB(1111),
where a1 = BB(1111) ↑↑↑↑ BB(1111),
an = B(1111) ↑a(n)-1 BB(1111)
Basically Graham's number, substituting each 3 with BB(1111), done with BB(1111) iterations rather than 64. After you've got that done, if you've got more time, write in some more ones, preferably in the first line.
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Anonymous2009-06-28 21:07
>>27
It's just trying to gauge where students are at in terms of knowledge and creativity/intelligence.
In most classes you'll play this with, all the answers suck. I've subbed both high school and community college math courses. The best answer I can remember getting was "10^10^10^10..." repeated until he ran out of space.
Basically, anyone who can use Knuth's up-arrow notation "wins". Anyone who has heard of more complicated numbers beats them and anyone who has heard of Busy Beavers beats everyone else. If two people use Busy Beavers without an obvious winner, they "tie".
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Anonymous2009-06-28 21:18
>>28
>While nested Busy Beavers seems a natural extension...
Wrong.
BB(x) grows faster than any computable function. Therefore, since BB(x) ↑↑↑↑ BB(x) is computable (once you know BB(x)), BB(BB(x)) gets bigger faster than BB(x) ↑↑↑↑ BB(x).
What you could do, is go nuts and do something like
You can probably do more than that even, but you get the idea. I don't know enough about the BB() function to know whether it's possible in principle to actually find it's value given infinite time, though.
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Anonymous2009-06-29 12:30
>>30
I think he was trying to make the point that there's nothing to cite for doing such a thing. That you're essentially "making something up". Original research would be against the rules. However, Busy Beavers HAVE been nested, by Stephen Kleene in 1943.
BB(BB(n)) = BB2(n)
Then of course you could do stuff like BBBB(y)(x).
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Anonymous2009-06-29 13:33
>>31 However, Busy Beavers HAVE been nested, by Stephen Kleene in 1943.
Are you sure? This is the first time I've heard of them to be honest, but the wikipedia article says the functions themselves were introduced in 1962.
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Anonymous2009-06-30 2:11
>>32
Ah, sorry, brain fart. We were talking about this in /g/.
Kleene published a paper called "Recursive predicates and quantifiers" in 1943, which was about some related concepts. His paper was then used in book called "The Undecidable" in 1965 which then used it to demonstrate nested Busy Beavers.
... eh, supposedly. I haven't actually read either. The whole thing is pointless mathematical masturbation, anyway. Who gives a shit?
I thought the point of BB was that if you ran a program on a computer of complexity N, and it didn't stop after BB(N) steps, then it would never stop. Therefore you could write a program to find a counterexample to Goldbach's conjecture, and if it didn't find one after Y billion years, then you'd know there wasn't any.
Or something like that.
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4tran2009-06-30 22:05
>>35
Except solving goldbach's conjecture is probably easier than finding any particular busy beaver #.
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Anonymous2009-07-01 4:01
In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe occupies.
That's fuckin awesome.
>>37
Yeah, that's the case for pretty much every number that's been written in this thread so far. The googolplex is just about the smallest serious answer in this thread.