What is the largest number that you can express in a simple string of no more than 35 keyboard characters?
You may use words or mathematical notation, or a combination of both, but your number must be unambiguously exact, and the definition must stand alone. (You can't have "den0eng3's number plus one", for example!)
I confess my maths is a bit rusty, so if this gets off the ground we may need to appoint an expert arbitrator!
Posts: 275 | Location: Portsmouth, United Kingdom | Registered: 06-17-02
"Billion", etc will be interpreted as being standard American usage, (unless you can make it clear otherwise as part of your 35 characters).
Mathematical operators must be used in the standard way that is recognised by Excel 97 formulae, i.e. ( ) brackets ^ raise to the power / divide * multiply + add - subtract
06-28-02, 10:19 AM dogspit 9Googolplex^(Googolplex^Googolplex) (A googolplex is a 1 followed by a googol zeroes)
06-28-02, 10:37 AM Bibc14 how about a term we havnent thrown in yet,
Googolplex^(Googolplex^Googolplex!)
! is factorial: #! is #* (#-1)* (#-2)....2
i dont know if excel recoginizes it, but calculators do.
-chris
06-28-02, 10:47 AM den0eng3 Bibc14: I just checked, and in Excel x! is written FACT(x) I'm afraid you will have to re-submit your entry. By the way I'm already at a loss to decide who is winning! Experts, please!
06-28-02, 10:50 AM den0eng3 dogspit: I've just spotted that your entry isn't valid, either. You would need an asterisk after the 9, but that would take you over the limit.
06-28-02, 12:47 PM Bibc14 den0eng3, if the operators must be able to work in excel, then the written out forms of numbers, such as billion, or Googolplex would not work either, so if it is the largest number that you can produce in excell in 35 characters it would it be:
FACT(99999999999999999999999999999)
or are we allowing written out numbers, and only excell operators.
06-28-02, 01:56 PM den0eng3
quote:Originally posted by Bibc14: are we allowing written out numbers, and only excell operators.
Yes. That was my intention. [Actually, written-out numbers would work in Excel if you defined them as names.]
Good luck, all. I'm off for a few days. I expect my mind to be boggled by the size of the numbers here when I get back!
06-28-02, 07:31 PM WiteoutKing If they're not defined, then use
FACT(9^9^9^9^9^9^9^9^9^9^9^9^9^9^9)
That's a lot bigger than 99999999999999999999999999!
06-28-02, 10:33 PM coldfuse Nice go of it, WOK! Can we find something larger?
06-29-02, 12:06 AM Mack Tuesday FACT(FACT(FACT(FACT(9^9^9^9^9^9))))
06-29-02, 12:18 AM Mack Tuesday Don't know if this will be accepted as legal, but here goes:
f(0)=9, f(n+1)=f(n)^f(n), f(9^9^99)
06-29-02, 12:27 AM Mack Tuesday By the way, here are the entries so far in order of increasing magnitude:
06-29-02, 09:37 PM WiteoutKing Would FACT(99^99^99^99^99^99^99^99^99^99) be larger than FACT(9^9^9^9^9^9^9^9^9^9^9^9^9^9^9) I'd think so... If so, then would it increase if you go FACT(999^999^999^999^999^999^99999) FACT(9999^9999^9999^9999^9999^9999) FACT(99999^99999^99999^99999^99999) FACT(999999^999999^999999^99999999) FACT(9999999^9999999^9999999999999) FACT(99999999^99999999^99999999999) FACT(999999999^999999999^999999999) etc.?
06-29-02, 10:28 PM coldfuse Are you heading backwards, towards FACT(99999...9), with no exponents at all? I'm glad you made your post, because it's good to toss everything around a bit, but this is my initial reaction.
Taking up three places with 9^9 produces a much larger number than 999. Taking up five spaces with 9^9^9 likewise produces a much larger number than 99999. I think you were on a better track to start with!
woops, just remembered another term excell uses E, as in 10E2 = 10 *10^2, if we are allowing that function, then:
fact(9^fact(9^fact(9^fact(9^9E9))))
06-30-02, 04:22 AM DrGerard First check the rules. If Excel is the reference, then the expression 3^3^3 means (3^3)^3, or 27^3 which is 19683, and not 3^(3^3) which is a 13-digit number. My own pocket calculator happens to hold the opposite view, but that's irrelevant...
Let's call Nn the largest number you can type with n keystrokes. The first values of Nn are tabulated below.
N1 9 N2 99 N3 9E9 has 10 digits. N4 9E99 has 100 digits. N5 9^9E9 has 8588182585 digits. N6 9^9E99 has over 8.588 1099 digits. N7 9^9E999 has over 8.588 10999 digits. N8 9^9E9999 has over 8.588 109999 digits. N9 9^(9^9E9) has over 0.95424 N5 digits. N10 9^(9^9E99) has over 0.95424 N6 digits. N11 9^(9^9E999) has over 0.95424 N7 digits. N12 9^EXP(9^9E9) has over 0.95424 exp(N5) digits. N13 9^EXP(9^9E99) has over 0.95424 exp(N6) digits.
Things start to get interesting with N12. Everybody's favorite (the FACT function) cannot come into play because its name is still too long, but EXP is almost as good and wins the day because of a shorter name.
For N13 we have to choose the larger of 9^EXP(9^9E99), 9^FACT(9^9E9), and 9^(9^(9^9E9))... Surprise! FACT is only the runner up, EXP wins by a long shot! The same thing will always happen thereafter because of the fast growth of the N sequence which allow us to feed only a "minuscule" argument to FACT compared to what can be fed to EXP because of the extra keystroke.
For 12 keystrokes or more, the expression for N is thus simply "9^EXP(...)" with an inner expression found 7 steps before in the table (7 fewer keystrokes). This makes the table extremely easy to extend beyond the 12th entry, and we may quickly obtain the final answer to the original question, as shown... 9^EXP(9^EXP(9^EXP(9^EXP(9^9E999)))) Functions exists which could help break that record (CH is slightly better than SH) but they have longer names in Excel (COSH and SINH) which disqualify them. Unless I missed an obscure Excel function with a short name and an exponential growth, you can't do better than this with strict Excel syntax. Can you?
[This message was edited by DrGerard on 06-30-02 at 04:31 AM.]
06-30-02, 04:54 AM DrGerard Something weird is happening when posting tables (I have no excuse for the poor spelling). roll eyes Time's up the post is permanent...
06-30-02, 07:37 AM DrGerard Oops. It turns out that the Answer Pool software is not at fault for the poorly displayed table above. (I was guilty of ending every row in the table with [Code Edited to prevent more problems - DG] instead of [Code Edited to prevent more problems - DG], for the information of any administrator reading this who might have the power/inclination to edit the post now.)
Sorry.
This message has been edited. Last edited by: DorianGreyed,
Posts: 275 | Location: Portsmouth, United Kingdom | Registered: 06-17-02
DrGerard, you have beaten me again! +++++++++++++++++++ 07-01-02, 02:56 PM FlyingHellfish How about 9↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑9 where:
9↑9 = 9*9*9*9*9*9*9*9*9 9↑↑9 = 9↑9↑9↑9↑9↑9↑9↑9↑9 9↑↑↑9 = 9↑↑9↑↑9↑↑9.... well, you get the idea.
Is ↑ an acceptable character? 07-01-02, 09:44 PM WiteoutKing That might work, but you'd have to "create" that formula.
07-01-02, 11:17 PM DrGerard FlyingHellfish was clearly going after the "free format" record, WoK, not the "Excel syntax" convention.
The arrow notation FH describes is well established and should be accepted. It was introduced by Donald E. Knuth in 1976 and is well known for producing unbelievably large numbers. By the way, FH, you may or may not know that the ordinary caret character (^) is accepted as a replacement for the arrow in that context, so there is no need for a "special" character (although the arrow looks better in print): A single caret means exponentiation, just like a single arrow, so everything is fine.
There is no universally accepted convention that beats FH number. So he takes the cake for the unrestricted category and I claim the "Excel syntax" subcategory. wink
However, this is not to say that even more "efficient" conventions do not exist. For example, you could decide that p^^^^^p can be abbreviated p[4]p, p^^^^^p would be p[5]p, etc. This means that FH's winner is "only" 9[33]9. With this nonstandard notation, we would reach new peaks with things like:
9[9]9, 9[9[9]9]9, 9[9[9[9]9]9]9, 9[9[9[9[9]9]9]9]9, etc. That's not the end of it, of course, as you can invent a new notation to abbreviate terms in THAT sequence, then in the new sequence that results from whatever newest convention you introduce, and so forth. This is why the "contest" is a puerile one (as advertised) unless you stick with some pre-existing formal syntax: Excel or widely accepted math notation, take your pick. For either one of this two choices, the contest is now closed... smile
07-02-02, 08:50 AM FlyingHellfish And we still can't even begin to touch Graham's number smile
07-02-02, 11:56 AM den0eng3
quote:
Originally posted by DrGerard: Excel or widely accepted math notation ... for either one of this two choices, the contest is now closed...
DrGerard’s psychological tactics are very cunning. His densely argued and apparently authoritative "final answer" had me fooled for a while. But I have an idea that I'm pretty sure can beat it. However, before showing my hand I want to get my idea checked out by a kosher mathematician.
Watch this space...
07-02-02, 12:50 PM DrGerard Graham's Number (the largest number ever involved in a legitimate mathematical proof) is huge indeed, but it's reached in a few keystrokes at the second stage of the process outlined in my previous post. Even at the first stage (explicitly given with my dubious "square bracket" notation) 260 keystrokes or so are enough to write a bigger number...
This reminds me that John H. Conway and Richard K. Guy have been advocating a "chained arrow" notation to describe monsters like Graham's Number and beyond... Their notation has been duly published and seems unchallenged by any competing system, so it's probably just as acceptable as Knuth's arrow notation, which you used.
What I previously described as a[b]c, Conway and Guy write a®b®c. Not much of a difference at this first stage, except that the horizontal arrow suggests a possible extension of the notation in a mind-boggling way. Formally, here is the definition (ellipses "..." stand for any particular chain of arrows):
First we state that ...®a®1 simply means ...®a
Then, the expression a...x®y®(z+1) is defined according to the value of y as follows: If y = 1, it is simply a...x If y=2, it is a...x®(a...x)®z If y=3, it is a...x®(a...x®(a...x)®z)®z
and so on...
This recursive definition may not be easy to grasp, but it does generate unbelievably huge numbers with just a few keystrokes. Graham's Number is somewhere between 3®3®64®2 and 3®3®65®2.
To conclude, if this notation is considered standard (I think it probably should), I'd claim the title for the "unrestricted" category (formerly yours) with the following entry: 9®9®9®9®9®9®9®9®9®9®9®9®9®9®9®9®9®9 That's only until someone comes up with the idea of abbreviating THAT with a notation like 9®®17. Mercifully, this time, nobody has ever proposed such a thing in print (as far as I know), because this would be so utterly useless (given that even Graham's Number is comfortably described with the "chained arrow" notation itself). cool
07-02-02, 01:19 PM DrGerard I'd love to assess your idea, Den0eng3 (unless you think I'm not "kosher" enough for you frown ). Otherwise, I hope whoever vets it finds the idea worthy of being posted (whether it's right or wrong), so we can all know what it was. A "never mind" future post on your part would be quite frustrating, given your promising hype... smile
07-02-02, 01:27 PM FlyingHellfish Graham's Number (the largest number ever involved in a legitimate mathematical proof) is huge indeed, but it's reached in a few keystrokes at the second stage of the process outlined in my previous post. Even at the first stage (explicitly given with my dubious "square bracket" notation) 260 keystrokes or so are enough to write a bigger number...
I now can see how your bracketing notation would indeed yield a larger number-I didn't take the time to consider the scope that the "Gerard Symbol" took on smile
07-02-02, 01:54 PM den0eng3
quote:Originally posted by DrGerard: ... unless you think I'm not "kosher" enough for you frown
Doc, I hope you didn’t think I was questioning your abilities! Far from it. It is precisely because you are plainly an absolute wizz at maths that I wanted to get my idea checked out before I posted it. Otherwise I’m sure you will be able to use my idea and change it around to get an even bigger number and thus reclaim the champion’s crown for yourself! I expect you will be able to do that anyway, but I’d like to give it my best shot.
07-02-02, 03:58 PM den0eng3 I am suddenly wracked with doubt, and all the mathematicians I know seem to be away, but I’ll post my idea anyway:
What I had in mind was EVALUATE("9E"&REPT(9,9^EXP(9^9E9)))
EVALUATE(T) evaluates the text string T as if it were an expression
& is an operator that concatenates two strings of text
REPT(T,n) yields a string of text comprising n successive copies of T [Excel converts numbers to text where the context requires, so there is no need to enclose the first argument in quotes]
07-02-02, 04:04 PM den0eng3 Note on the EVALUATE function:
This is a “macro sheets only” Excel function. Since Microsoft are trying to discourage the use of macro sheets in favour of the [in my view] cumbersome and inelegant Visual Basic for Applications, the “macro sheets only” functions have not been documented in any Excel help files since version 5. However, macro sheets are still supported in all versions of Excel that I have seen, including Excel 97.
If you doubt my word, try this: In Excel, open a new workbook. Right-click on one of the sheet tabs, and select “Insert” from the short-cut menu. Then double-click on the “MS Excel 4.0 Macro” icon. In any cell in the new sheet, type =EVALUATE("1E6") Now change the settings from “display formulae” to “display values”. The Windows short-cut for this is Ctrl+` (` is just to the left of 1 on my UK-layout keyboard). It displays 1000000, does it not?
You might still argue that only standard worksheet functions should be allowed, in which case I would counter that the dictionary definition of the word "evaluate" is unambiguous – “to determine or fix the value of”, so my entry is still valid, under my first supplementary rule.
What do you reckon, Doc?
[This message was edited by den0eng3 on 07-02-02 at 04:35 PM.]
07-02-02 07:18 PM DrGerard Unfortunately, your own rules put the limit at 35 keystrokes, and this is what kills you. frown
The number you propose happens to be fairly small: For any number n, REPT(9,n) is only 10^n-1. Similarly, putting a string of m nines after "9E" using text concatenation only forms a string equivalent to 9*10^(10^m-1). In other words, your entry: EVALUATE("9E"&REPT(9,9^EXP(9^9E9))) is numerically equal to 9*10^(10^(10^9^EXP(9^9E9)-1)-1) Even this exact string (31 keystrokes) is larger than your proposal, which is thus very far from the 35-keystroke title.
However, an extension or your idea would take the cake for longer strings, because of either of the following patterns, where BOTH instances of ... represent the SAME properly formed number:
When the inner expression "..." is of length n, the first one is of length 2n+38 and the second is of length 2n+39, so you use either one depending on the parity of the number of keystrokes you are allowed. This method is already a winner with 40 or 41 keystrokes, and it's a definite killer for 42 strokes or more (let alone 118 keystrokes or more, where you can use it several times). With 39 keystrokes or less, it's of no help.
07-02-02, 07:46 PM DrGerard In my previous post, the 31-keystroke expression: 9*10^(10^(10^9^EXP(9^9E9)-1)-1) should be replaced by the 33-keystroke expression: 9*10^(10^(10^(9^EXP(9^9E9))-1)-1)
07-02-02, 08:34 PM DrGerard Because the following pattern adds only 35 characters, it allows one or two extra keystrokes in the two critical (equal) inner expressions represented by ellipses (...), it is superior to what I previously proposed and takes the cake for 39 keystroke or more. EVALUATE(REPT("9^(",...)&9&REPT(")",...)) For 37 or 38 keystrokes however, the thing becomes: EVALUATE(REPT("9^(",9)&9&REPT(")",9)) which is equivalent to 9^(9^(9^(9^(9^(9^(9^(9^(9^(9)))))))))) This happens to be itself a 37-keystroke expression much smaller than the best 37-keystroke or 38-keystrokes expressions, namely: 9^EXP(9^EXP(9^EXP(9^EXP(9^(9^9E9))))) 9^EXP(9^EXP(9^EXP(9^EXP(9^(9^9E99))))) Too bad this is just outside the "35 keystroke" scope!
07-03-02, 09:43 AM den0eng3 DrGerard, you are the man! smile
07-03-02, 12:14 PM FlyingHellfish
quote:Originally posted by den0eng3: DrGerard, you are the man! smile
I concur!
07-16-02, 12:41 PM WiteoutKing (throws in towel)
07-22-02, 09:11 PM WiteoutKing I just remembered something... Excel requires the = sign of anything that is not only numbers. So, 3EXP3 would show up as 3EXP3 instead of 27.
07-24-02, 09:41 PM XaurreauX A puerile big number contest Don't ask me, I flunked math in eighth grade. confused
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Posts: 1363 | Location: Lowell, MA, USA | Registered: 06-03-02
