Commas in a Googol Plex
Moshe Jacobson
have no idea how to solve.
How many digits appear before the first comma in a Googol Plex?
(Assuming every 3 digits from the right are separated by a comma, and
assuming that a googol plex is a 1 followed by a googol zeros, and a
googol is 1 followed by 100 zeros.)
Obviously this question is asking for the modulus 3 of a googol one.
Moshe
--
*** SPAM BLOCK: Remove bra before replying! ***
http://runslinux.net :: moshe at runslinux dot net :: AIM: Jehsom
Mark Brader
> A few days ago, I was presented with the following question, which I
> have no idea how to solve.
>
> How many digits appear before the first comma in a Googol Plex?
> (Assuming every 3 digits from the right are separated by a comma, and
> assuming that a googol plex is a 1 followed by a googol zeros, and a
Spoiler:
40
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3
Simple. 10 is congruent to 1 mod 3. Therefore 10 to the power N is
congruent to 1 to the power N, which is 1, for any positive integer N.
Therefore a googol = 10^100 is congruent to 1 mod 3. A googolplex
has googol+1 digits, so the number of digits is congruent to 2 mod 3.
So there are 2 digits before the first comma.
--
Mark Brader, Toronto "As for Canada's lack of mystique,
m...@vex.net it is not unique." -- Mark Leeper
My text in this article is in the public domain.
Michael Mendelsohn
Another way to put it is that if you compute 10^N - 1, you get a number
with all digits 9, and that is divisible by 3; so all numbers of the
form 10^N have a remainder of 1 when divided by 3.
> Therefore a googol = 10^100 is congruent to 1 mod 3. A googolplex
> has googol+1 digits, so the number of digits is congruent to 2 mod 3.
> So there are 2 digits before the first comma.
A googolplex = 10,000,.....
Mark, you have beaten me to the solution yet again! :)
Cheers
Michael
--
Still an attentive ear he lent Her speech hath caused this pain
But could not fathom what she meant Easier I count it to explain
She was not deep, nor eloquent. The jargon of the howling main
-- from Lewis Carroll: The Three Usenet Trolls
Mark Brader
> Mark, you have beaten me to the solution yet again! :)
Well, somebody had to do it! :-)
--
Mark Brader, Toronto | "He is even more important than my cat,
m...@vex.net | which is saying something." --Flash Wilson
Kannappan
> has googol+1 digits, so the number of digits is congruent to 2 mod 3.
> So there are 2 digits before the first comma.
Yeah Exactly. Moreover the answer is always the same for all the below
10^10
10^100
10^1000
10^10000
10^100000
and so on....
In effect for 10^10^N where N is any whole number
there are always 2 digits 1 and 0 before the comma.
Thanks
Kannappan
http://www.gulmohar.blogspot.com
Ed Murphy
> A few days ago, I was presented with the following question, which I have
> no idea how to solve.
>
> How many digits appear before the first comma in a Googol Plex? (Assuming
> every 3 digits from the right are separated by a comma, and assuming that
> a googol plex is a 1 followed by a googol zeros, and a googol is 1
> followed by 100 zeros.)
>
> Obviously this question is asking for the modulus 3 of a googol one.
It's fairly well-known that the modulus 3 of N equals the modulus 3 of
the sum of the digits of N. The digits of a google one are 1 + lots of
0's + 1, which sum to 2.
Michael Mendelsohn
Just when you thought the solution couldn't become any simpler,
it suddenly does.
:-)
Moshe Jacobson
> It's fairly well-known that the modulus 3 of N equals the modulus 3
> of the sum of the digits of N. The digits of a google one are 1 +
> lots of 0's + 1, which sum to 2.
I presented this problem to my coworker after it was solved here with
the previous method, and he used exactly this method. I have a smart
coworker :-)
Rich Grise
>
> A googolplex = 10,000,.....
>
Heh - Carl Sagan once started unrolling a paper tape (like a cash register
tape) with the beginning of a googleplex on it in marking pen, and he
started walking down the street unrolling this tape, and across town, and
all over the place - through the park, etc, for dramatic effect of course
- he runs out of tape, and says, "A piece of paper big enough to write the
number 'one googleplex' could not be _stuffed_ into the _known Universe_!"
emphasis his, of course. ;-)
Cheers!
Rich
Patrick Hamlyn
Ah the miracle of the internet. Impossible on paper, yet here you've done it
three times in one post.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms...@egroups.com)
Bill Smythe
> .... Carl Sagan ....
> .... says, "A piece of paper big enough to write the
> number 'one googleplex' could not be _stuffed_ into the _known Universe_!"
Not even in billions and billions of years?
Bill Smythe
Michael Mendelsohn
Not even in billions to the power of billions years, I think.
Rich Grise
> Bill Smythe schrieb:
>> "Rich Grise" wrote:
>> > .... Carl Sagan ....
>> > .... says, "A piece of paper big enough to write the
>> > number 'one googleplex' could not be _stuffed_ into the _known
>> > Universe_!"
>>
>> Not even in billions and billions of years?
>
> Not even in billions to the power of billions years, I think.
Just in case you're serious, "Billions and billions" is one of the most
famous things he never said, except once, in an interview, just kind of
like, "Aw, ok, billions and billions, ha ha" when the interviewer asked
him about it. He did say, however, that he did emphasize the "B" sound,
so that people wouldn't think he was talking about mere millions. :-)
Cheers!
Rich
Bill Smythe
> Just in case you're serious, "Billions and billions" is one of the most
I distinctly remember him saying "billions and billions" (probably just
once) in his PBS series, probably the first episode, several years ago. At
that time he was not trying to make fun of himself, either.
Bill Smythe
Mark Brader
>> Just in case you're serious, "Billions and billions" is one of the most
>> famous things he never said ....
He (Carl Sagan, for those just tuning in) did use it as a book title
after it became a catchphrase, though.
Bill Smythe:
> I distinctly remember him saying "billions and billions" (probably just
> once) in his PBS series, probably the first episode, several years ago.
Why would you distinctly remember it when it wasn't a catchphrase yet?
It became one after Johnny Carson used it in a parody of the show.
--
Mark Brader, Toronto "These Millennia are like buses."
m...@vex.net --Arwel Parry
Bill Smythe
> Why would you distinctly remember it when it wasn't a catchphrase yet?
Probably for the very reason it became a catchphrase -- it was EXTREMELY
unforgettable, especially with his strong accent on the first syllable
(especially the letter B) in BOTH of the "billions"s.
Bill Smythe
Rich Grise
I'd almost bet you're remembering Johnny Carson saying it, and your memory
is helping out a bit. :-)
http://en.wikipedia.org/wiki/Carl_Sagan
about 1/3 of the way down the page, under "Popularization of science".
I hope you know how much I love to say, "I told you so!" ;-p
Cheers!
Rich
Mark Brader
> http://en.wikipedia.org/wiki/Carl_Sagan
> about 1/3 of the way down the page, under "Popularization of science".
>
> I hope you know how much I love to say, "I told you so!" ;-p
I hope you don't think that's an authoritative source. Most likely
whoever wrote that bit was simply paraphrasing what Sagan wrote in
the book, rather than having checked it independently. Now I'm not
saying that Sagan was wrong, but the previous poster *is*.
--
Mark Brader, Toronto | "...blind faith can ruin the eyesight--
m...@vex.net | and the perspective." --Robert Ludlum
Bill Smythe
> I'd almost bet you're remembering Johnny Carson saying it, and your memory
> is helping out a bit. :-)
I do indeed remember Johnny Carson saying it, but I also remember
remembering Carl Sagan saying it when I heard Johnny Carson say it (if
anybody can follow that).
Carl Sagan's TV series was available on VHS at one time (I saw it in a
bookstore). If anybody out there has access to those tapes, maybe you can
help out by watching the first episode or two.
Bill Smythe
r.e.s.
If this is about whether Sagan used the expression "billions & billions"
on PBS, I can say that I distinctly recall him doing so. (I didn't see
the Carson show.)
--r.e.s.
Earle Jones
"Bill Smythe" <chic...@beforeRCNafter.com> wrote:
*
One thing Carl Sagan did say is this:
"The idea that God is an oversized white male with a flowing
beard who sits in the sky and tallies the fall of every sparrow is
ludicrous. But if by "God" one means the set of physical laws that
govern the universe, then clearly there is such a God. This God is
emotionally unsatisfying...it does not make sense to pray to the law
of gravity."
--Carl Sagan
earle
*
Phil Carmody
> One thing Carl Sagan did say is this:
>
> "The idea that God is an oversized white male with a flowing
> beard who sits in the sky and tallies the fall of every sparrow is
> ludicrous. But if by "God" one means the set of physical laws that
> govern the universe, then clearly there is such a God. This God is
> emotionally unsatisfying...it does not make sense to pray to the law
> of gravity."
>
> --Carl Sagan
Too long for a .sig, alas. Unlike this one:
If you want to save your child from polio, you can pray or you can inoculate.
-- Carl Sagan (1934-1996), The Demon-Haunted World, p. 204 p. 30
Phil
--
I used to have an interest in writing viral code and lost interest
quickly when Win95 came out. Hell how could any of us in the scene
write a more invasive program than Win95. It made us all obsolete.
-- Screaming Radish [NuKE] on alt.comp.virus.source.code
Earle Jones
Phil Carmody <thefatphi...@yahoo.co.uk> wrote:
> Earle Jones <earle...@comcast.net> writes:
> > One thing Carl Sagan did say is this:
> >
> > "The idea that God is an oversized white male with a flowing
> > beard who sits in the sky and tallies the fall of every sparrow is
> > ludicrous. But if by "God" one means the set of physical laws that
> > govern the universe, then clearly there is such a God. This God is
> > emotionally unsatisfying...it does not make sense to pray to the law
> > of gravity."
> >
> > --Carl Sagan
>
> Too long for a .sig, alas. Unlike this one:
>
> If you want to save your child from polio, you can pray or you can inoculate.
> -- Carl Sagan (1934-1996), The Demon-Haunted World, p. 204 p. 30
>
> Phil
*
Or this:
"Faith" is a fine invention
When Gentleman can see-
But Microscopes are prudent
In an Emergency."
--Emily Dickinson
earle
*
Trog Woolley
In his book Cosmos he writes "A galaxy is composed of gas and dust and stars -
billions upon billions of stars. Every star may be a sun to someone".
This is in the first chapter "The Shores Of The Cosmic Ocean". He doesn't
use the words "billions and billions" in the first episode of the series Cosmos,
but I'll bet he does somewhere, as most of the text of the book appears in it.
--
Trog Woolley | trog at trogwoolley dot com
(A Croweater back residing in Pommie Land with Linux)
Isis Astarte Diana Hecate Demeter Kali Inanna
Surendar Jeyadev
>bookstore). If anybody out there has access to those tapes, maybe you can
>help out by watching the first episode or two.
Good Lord! Why?? Wasn't once awful enough?
--
Surendar Jeyadev jeya...@wrc.xerox.com
Remove 1 for email address
Gerry Quinn
thefatphi...@yahoo.co.uk says...
> Too long for a .sig, alas. Unlike this one:
>
> If you want to save your child from polio, you can pray or you can inoculate.
> -- Carl Sagan (1934-1996), The Demon-Haunted World, p. 204 p. 30
Before choosing that, perhaps one should wait and see how research into
the origins of AIDS, and the epidemiology of the disease in more or less
religious regions, pan out.
- Gerry Quinn
Jules d'Entremont
>
> If this is about whether Sagan used the expression "billions & billions"
> on PBS, I can say that I distinctly recall him doing so. (I didn't see
> the Carson show.)
>
> --r.e.s.
Sagan's book "Billions and Billions" starts off with the following:
"I never said it. Honest. Oh, I said there are maybe 100 billion
galaxies and 10 billion trillion stars. It's hard to talk about the
Cosmos without using big numbers. I said "billion" many times on the
Cosmos television series, which was seen by a great many people. But I
never said "billions and billions." For one thing, it's too imprecise.
How many billions are "billions and billions"? A few billion? Twenty
billion? A hundred billion? "Billions and billions" is pretty vague.
When we reconfigured and updated the series, I checked - and sure
enough, I never said it.
But Johnny Carson - on whose Tonight Show I'd appeared almost thirty
times over the years - said it."
For the rest, get the book.
Jules.
--
Jules d'Entremont ju...@dentremont.com
Nepean, Ontario
r.e.s.
> r.e.s. wrote:
>>
>> If this is about whether Sagan used the expression "billions & billions"
>> on PBS, I can say that I distinctly recall him doing so. (I didn't see
>> the Carson show.)
>>
>
> "I never said it. Honest. Oh, I said there are maybe 100 billion
> galaxies and 10 billion trillion stars. It's hard to talk about the
> Cosmos without using big numbers. I said "billion" many times on the
> Cosmos television series, which was seen by a great many people. But I
> never said "billions and billions." For one thing, it's too imprecise.
> How many billions are "billions and billions"? A few billion? Twenty
> billion? A hundred billion? "Billions and billions" is pretty vague.
> When we reconfigured and updated the series, I checked - and sure
> enough, I never said it.
> But Johnny Carson - on whose Tonight Show I'd appeared almost thirty
> times over the years - said it."
>
> For the rest, get the book.
I'll take your word (and his ;o) that he didn't make the remark in
the Cosmos series -- but the question is whether he said it on PBS,
not necessarily in Cosmos. One possibility, suggested by the fact
that others also recall it in connection with Cosmos, is that there
might have been an ad for Cosmos that contained footage not shown
in the series itself.
--r.e.s.
Bill Smythe
> ".... When we reconfigured and updated the series, I checked - and sure
> enough, I never said it. ...."
By any chance, was the checking done after the reconfiguring, and if so,
could the phrase have been included in the old, un-reconfigured version?
Also, one poster here speculated that he might have said "billions upon
billions" rather than "and".
Bill Smythe
Phil Carmody
> r.e.s. wrote:
>
> >
> > If this is about whether Sagan used the expression "billions & billions"
> > on PBS, I can say that I distinctly recall him doing so. (I didn't see
> > the Carson show.)
> >
> > --r.e.s.
>
> Sagan's book "Billions and Billions" starts off with the following:
>
> "I never said it. Honest. Oh, I said there are maybe 100 billion
> galaxies and 10 billion trillion stars. It's hard to talk about the
> Cosmos without using big numbers. I said "billion" many times on the
> Cosmos television series, which was seen by a great many people. But
> I never said "billions and billions."
Side issue - didn't we have a citation for "billions of billions",
though, only a month or so back?
> For one thing, it's too
> imprecise. How many billions are "billions and billions"? A few
> billion? Twenty billion? A hundred billion? "Billions and billions"
> is pretty vague.
This is coming from the person who devotes two whole full near-A4 pages
in /Cosmos/ to the freaking Drake equation?????
ObPuzzle:
Why are the three letters "A", "B", and "H" at the forefront of my mind
presently?
Phil
--
The gun is good. The penis is evil... Go forth and kill.
Chuck Grant
Phil Carmody wrote:
> ObPuzzle:
> Why are the three letters "A", "B", and "H" at the forefront of my mind
> presently?
>
>
> Phil
That's really more of a triva question than a puzzle.
Settle down Bevis.
Chuck Grant
Phil Carmody
> Phil Carmody wrote:
>
> > ObPuzzle:
> > Why are the three letters "A", "B", and "H" at the forefront of my mind
> > presently?
> > Phil
>
If I'd have said "B", "H", and "A", it would have been a trivia
question, with an immediate google solution. The puzzle was
noticing that the order was not specified.
;-)
George Weinberg
<thefatphi...@yahoo.co.uk> wrote:
Are they terms in the freaking Drake equation?
George
Phil Carmody
> >This is coming from the person who devotes two whole full near-A4 pages
> >in /Cosmos/ to the freaking Drake equation?????
> >
> >
> >ObPuzzle:
> >Why are the three letters "A", "B", and "H" at the forefront of my mind
> >presently?
> >
> >
> >Phil
>
> Are they terms in the freaking Drake equation?
Yes! Well done! For those who aren't familiar with the
freaking Drake equation, here's what they stand for:
A is the Arbitrariness adjustment
B is the Bullshit factor
H is the Handwaving coefficient
ephramb...@gmail.com
> A few days ago, I was presented with the following question, which I
> have no idea how to solve.
>
> How many digits appear before the first comma in a Googol Plex?
> (Assuming every 3 digits from the right are separated by a comma, and
> assuming that a googol plex is a 1 followed by a googol zeros, and a
> googol is 1 followed by 100 zeros.)
>
> Obviously this question is asking for the modulus 3 of a googol one.
>
> Moshe
>
> --
> *** SPAM BLOCK: Remove bra before replying! ***
> http://runslinux.net :: moshe at runslinux dot net :: AIM: Jehsom
Richard Heathfield
> On Monday, 25 October 2004 20:59:34 UTC+1, Moshe Jacobson wrote:
>> A few days ago, I was presented with the following question, which I
>> have no idea how to solve.
>>
>> How many digits appear before the first comma in a Googol Plex?
>> (Assuming every 3 digits from the right are separated by a comma, and
>> assuming that a googol plex is a 1 followed by a googol zeros, and a
>> googol is 1 followed by 100 zeros.)
>>
>> Obviously this question is asking for the modulus 3 of a googol one.
Still, never mind that. Nowadays, one takes one's traffic where one can
find it.
Okay. If there is an exact multiple-of-3 zeros after the 1, there would
be one digit before the comma (1,000 or 1,000,000 or 1,000,000,000 etc).
If there is one more than an exact multiple-of-3 zeros after the 1,
there would be two digits before the comma (10,000 or 10,000,000 or
10,000,000,000 etc).
If there are two more than an exact multiple-of-3 zeros after the 1,
there would be three digits before the comma (100,000 or 100,000,000 or
100,000,000,000 etc).
If there are three more, we go back to one digit before the comma
(because if N is a multiple of 3, so is N+3).
The digital root of an integer is the sum of its digits, repeating as
necessary until a single digit remains.
An integer is evenly divisible by 3 if and only if its digital root is
evenly divisible by 3 - i.e. is 3, 6, or 9.
The digital root of a googol (G) is 1 + (tons of zeros), so it's 1.
The digital root of (G + 2) --- that is, 1000000...0000002 --- is
therefore 3. So if the question had been "how many digits before the
first comma of a 1 followed by G+2 zeros?", the answer would clearly be
1. Let's add a zero, thus changing the comma pattern.
How many digits before the first comma of a 1 followed by G+3 zeros?
Clearly the answer is 2, because we've added a digit.
We can now eliminate our extra group of three zeros without changing the
pattern of commas, so the answer is still 2.
Quod Erat DemonperhapsatriflewordilybutIthinkcorrectlystrandum.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within
Gene Wirchenko
wrote:
>On Monday, 25 October 2004 20:59:34 UTC+1, Moshe Jacobson wrote:
>> A few days ago, I was presented with the following question, which I
>> have no idea how to solve.
>>
>> How many digits appear before the first comma in a Googol Plex?
>> (Assuming every 3 digits from the right are separated by a comma, and
>> assuming that a googol plex is a 1 followed by a googol zeros, and a
>> googol is 1 followed by 100 zeros.)
>>
>> Obviously this question is asking for the modulus 3 of a googol one.
This means that the remainder of a googol / 3 = 1. That means that
the bit before the first comma is 10 so the answer is 2.
Sincerely,
Gene Wirchekno