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BNC
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A challenge
« on: Oct 25th, 2003, 9:24am » |
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Here is a challenge (or a competition if you wish...)
Create the largest number using:
1. The digits 1,2,and 3 once only.
2. Any mathematical symbols you want, but no symbol is to be used more than once.
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Sir Col
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Re: A challenge
« Reply #1 on: Oct 25th, 2003, 10:05am » |
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Is there a requirement for the value to be finite? It not, how about w(3^21)!?
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BNC
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Re: A challenge
« Reply #2 on: Oct 25th, 2003, 10:29am » |
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OK, let's make it a finite number.
Although it doesn't matter -- we could use w<a large finite>
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Icarus
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Re: A challenge
« Reply #3 on: Oct 25th, 2003, 12:38pm » |
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[theta](321)
Where [theta] gets to be whatever I want to definite it as! (Show me your competing value, and I'll tell you what [theta] is to beat it!) 
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Sir Col
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Re: A challenge
« Reply #4 on: Oct 25th, 2003, 3:33pm » |
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In which case, I define the dynamic function, [chi](n), to be the sum of the absolute value of all existing finite functions, and any functions that are ever defined, excluding itself.
Whatever value you compute, n, I will feed into my function: for example, [chi](n)=n! + [sqrt]n + [theta](n) + ...
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Icarus
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Re: A challenge
« Reply #5 on: Oct 25th, 2003, 4:58pm » |
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Aren't you forgetting something?
on Oct 25th, 2003, 10:29am, BNC wrote:| OK, let's make it a finite number. |
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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SWF
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Re: A challenge
« Reply #6 on: Oct 25th, 2003, 7:41pm » |
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This could be a pretty good riddle if the list of functions allowed is more clearly defined (and restricted). Why not put this in one of the riddle forums? Also do tanh and cosh count as using different symbols or is that two h's?
Here is a quick attempt using some of the more common symbols and functions (I guess the various brackets and parentheses are not needed, but for clarity they were included, being careful to avoid repetition. ):
[Gamma]|[lceil]2/[1-[surd]tanh[exp{cosh(sinh 3)}][rceil]!|
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BNC
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Re: A challenge
« Reply #7 on: Oct 27th, 2003, 4:10am » |
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You know, I though leaving it "open-ended" may actually help achieving more interesting posts. But seeing how it was misused let me suggest a few rules:
1. Use only a single-character symbols. So [sqrt] is ok, but sin() is not. e is ok, but exp is not (yeah, I know…). Parenthesis are an allowed exception.
2. Use only known symbols. How can we define "known"? I suggest it meets at least one of these criteria:
- At least 5 web pages have content using that symbol
- It appears on a mathematical encyclopedia
- It's implemented as a standard function in one of the common mathematical tools.
So "+" is ok, but [theta] is not. "!" is ok, but [chi] is not.
What do you say?
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towr
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Re: A challenge
« Reply #8 on: Oct 27th, 2003, 4:19am » |
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on Oct 27th, 2003, 4:10am, BNC wrote:| - At least 5 web pages have content using that symbol |
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Let's make that five webpages made by someone you don't know 
Else someone might just make them himself, or get someone else to do it..
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Sir Col
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Re: A challenge
« Reply #9 on: Oct 27th, 2003, 5:31am » |
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on Oct 27th, 2003, 4:10am, BNC wrote:
I've found it interesting so far, but it seems reasonable.
How about, e^pi^((3^21)!) ?
Am I allowed to use repeated exponents? In my attempt I've used the ^ symbol more than once, but that is only because I cannot use the [sup] tag more than once.
What about placing the given digits together to form another number; e.g., 321?
What about banishing mathematical constants too? I think that would make it fairer, as my knowledge of them is limited to the basic ones.
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| « Last Edit: Oct 27th, 2003, 5:33am by Sir Col » |
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BNC
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Re: A challenge
« Reply #10 on: Oct 27th, 2003, 7:54am » |
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I'm only suggesting, and am always open for other opnions.
on Oct 27th, 2003, 5:31am, Sir Col wrote:
Am I allowed to use repeated exponents?
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I suggest limiting ourselvs to a single exponent.
Quote:
What about placing the given digits together to form another number; e.g., 321?
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Yes, sure.
Quote:
What about banishing mathematical constants too?
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On 2nd thought, this is a good idea. After all, constants are numbers, and we're limitted to 1,2,3 (see, I told you I'm open to suggestions;))
Here's one try of my own...
(.1^-32)!
Of course, it leaves the question of how to derermine what number is larger out of two really large ones...
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towr
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Re: A challenge
« Reply #11 on: Oct 27th, 2003, 8:27am » |
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on Oct 27th, 2003, 7:54am, BNC wrote:| Here's one try of my own... |
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That's easy to improve upon
:: ((1%)^-32)! ::
hmm, if I were allowed to use CS functions, ~1 would be a good starting point (~ is bitwise negation, so binary 0000000000000001, would become 111111111111110. Only problem is we don't have a max_int here)
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| « Last Edit: Oct 27th, 2003, 8:28am by towr » |
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Sir Col
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Re: A challenge
« Reply #12 on: Oct 27th, 2003, 8:35am » |
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Cor, now that's big! I don't think we're going to beat it. 
What a clever idea BNC; and nice improvement, towr.
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BNC
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Re: A challenge
« Reply #13 on: Oct 27th, 2003, 9:07am » |
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on Oct 27th, 2003, 8:27am, towr wrote:
That's easy to improve upon
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As does your (by combining both ideas) to get ::[(.1%)^-32]!::
In fact, if we were to use paper, we could use the [promil] symbol (O/OO) to generate .1[promil]
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Sir Col
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Re: A challenge
« Reply #14 on: Oct 27th, 2003, 9:10am » |
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on Oct 27th, 2003, 8:35am, Sir Col wrote:| I don't think we're going to beat it. |
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Why don't I just keep my posts to a minimum! 
What is the promil symbol?
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BNC
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Re: A challenge
« Reply #15 on: Oct 27th, 2003, 9:35am » |
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on Oct 27th, 2003, 9:10am, Sir Col wrote:
What is the promil symbol? |
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I'm sorry, it seems in English it doesn't have a name -- it's just called "one thousandth". Its symbol is similar to "percent", but with two "circles" innstead of one below the "/".
If you have access to "M$-word", it should be in "insert symbol" (at least it's on my version) quite down the page, just above the 1/3, 2/3... symbols.
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Sir Col
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Re: A challenge
« Reply #16 on: Oct 27th, 2003, 10:30am » |
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Thanks, BNC. I couldn't find it in Character map, but I discovered that it is a standard HTML web character: ‰ = ‰.
If you ever want to display a HTML/ASCII entity, and you know the decimal code, you can hold down ALT and type 0 followed by its 3-digit code. For example, I often use ALT+0150 = –, as better minus sign in my posts.
I've noticed you say before that English is not your first language; if you don't mind me asking, what is your language?
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| « Last Edit: Oct 27th, 2003, 10:32am by Sir Col » |
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Barukh
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Re: A challenge
« Reply #17 on: Oct 27th, 2003, 11:41am » |
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Here's my attempt to bit towr's number: e^321!.
Let's take logarithm of both towr's number and mine, and use a very rough approximation ln(n!) ~ n ln(n). Then:
ln((100^32)!) ~ 100^32 ln(100^32) = 147 * 100^32. This number consists of 67 digits.
ln(e^321!) = 321!. This number consists of ~800 digits.
Does it make any sense?!
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Sir Col
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Re: A challenge
« Reply #18 on: Oct 27th, 2003, 12:58pm » |
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I think that BNC decied that the use of mathematical constants were not allowed, as you are effectively using a number other than 1, 2, and 3.
Don't forget that BNC found a slightly improved version: (.1‰)-32 = 1x10128 (129 digits) .
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SWF
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Re: A challenge
« Reply #19 on: Oct 27th, 2003, 6:07pm » |
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To find the biggest number possible would be quite a research project to figure out every function that can be used. Since nesting tanh, cosh, sinh, and exp is not allowed, here are a few more suggestions for rapidly growing functions that fit the rules:
[gamma](n)= the gamma function (supposed to be a capital gamma, but couldn't make one)- for integer n equals (n-1)!
n! = n factorial
[lfloor]n = n factorial (this is an older notation which I did not display properly. It looks as if you drew a square around n, then erased the top and right sides.
[sum](n)= the factorial sum function= 1! + 2! + 3! + ... n!
NM= "tetration" also known as power tower= M^M^M^...M (the number of M's is N).
I haven't figured out the best order, (why bother when I know there are still other functions that can further raise the number), but you can get something at least as big as 4 nested factorials, and then power tower it with 2 or 3 to get a giant number.
For example x2, where x is the biggest number you can get using just the 1 and 3, maybe something like:
x=[sum]([gamma]([lfloor](3/.1% ‰!)))
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Icarus
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Re: A challenge
« Reply #20 on: Oct 27th, 2003, 6:31pm » |
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on Oct 27th, 2003, 12:58pm, Sir Col wrote:| I think that BNC decied that the use of mathematical constants were not allowed, as you are effectively using a number other than 1, 2, and 3. |
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on Oct 27th, 2003, 4:10am, BNC wrote:| 1. Use only a single-character symbols. So [sqrt] is ok, but sin() is not. e is ok, but exp is not (yeah, I know…). Parenthesis are an allowed exception. |
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on Oct 27th, 2003, 4:10am, BNC wrote:2. Use only known symbols. How can we define "known"? I suggest it meets at least one of these criteria:
- At least 5 web pages have content using that symbol
- It appears on a mathematical encyclopedia
- It's implemented as a standard function in one of the common mathematical tools.
So "+" is ok, but [theta] is not. "!" is ok, but [chi] is not. |
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I should point out that both [theta] and [chi] are used to represent standard (but high level, particularly for [theta]) functions.
So what you are ruling out here are not the standard versions, but the facetious definitions that I and Sir Col gave.
That said, I think a little research should turn up means to dwarf even Barukh's monster. In particular, Ackerman functions ought to do it. And some of the expansions, such as the one someone linked to in a thread recently (but I am of course too lazy to go look up right now). You just have to find a notation for them that meets BNC's conditions.
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BNC
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Re: A challenge
« Reply #21 on: Oct 27th, 2003, 11:44pm » |
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Hey Icarus,
quating Sir Col and me out of order to make it look like a contrudiction is really unfair! 
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towr
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Re: A challenge
« Reply #22 on: Oct 28th, 2003, 12:49am » |
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on Oct 27th, 2003, 11:44pm, BNC wrote:| quoting Sir Col and me out of order to make it look like a contrudiction is really unfair! |
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Yes I agree, so here's the proper order. While BNC first allowed e, he reconsidered, since it's a constant (and thus another number)
on Oct 27th, 2003, 4:10am, BNC wrote:| 1. Use only a single-character symbols. So [sqrt] is ok, but sin() is not. e is ok, but exp is not (yeah, I know…). Parenthesis are an allowed exception. |
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on Oct 27th, 2003, 5:31am, Sir Col wrote:| What about banishing mathematical constants too? |
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on Oct 27th, 2003, 7:54am, BNC wrote:| On 2nd thought, this is a good idea. After all, constants are numbers, and we're limitted to 1,2,3 (see, I told you I'm open to suggestions;)) |
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Sir Col
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Re: A challenge
« Reply #23 on: Oct 28th, 2003, 7:46am » |
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Something that SWF suggested offers an improvement on the .1 variant: ((.1%)‰)-32 = 10192 (193 digits); then using factorial makes it massive.
Are allowing functions: gamma, tetration, euler-phi, and so on; or are we restricting it to basic operators: factorial, roots, arithmetic, and so on?
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BNC
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Re: A challenge
« Reply #24 on: Oct 28th, 2003, 7:58am » |
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on Oct 28th, 2003, 7:46am, Sir Col wrote:| Are allowing functions: gamma, tetration, euler-phi, and so on; or are we restricting it to basic operators: factorial, roots, arithmetic, and so on? |
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I'm not sure. What do you think?
I seems to me that the "basic core" is more important, as we can always "plug in" the other functions. But it sure will be more interesting to see what weird functions people may dig up!
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