Internet Info 1994 March/Internet Info CD-ROM (Walnut Creek) (March 1994).iso/answers/rec/puzzles/archive/pickover/part3

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From: [e-mail] (Chris Cole)
Subject: rec.puzzles Archive (pickover), part 30 of 35
Message-ID: <puzzles/archive/pickover/[e-mail]>
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Summary: This is part of an archive of questions
 and answers that may be of interest to
 puzzle enthusiasts.
 Part 1 contains the index to the archive.
 Read the rec.puzzles FAQ for more information.
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Archive-name: puzzles/archive/pickover/part3
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Version: 4


==> pickover/pickover.12.p <==
Title: Cliff Puzzle 12: Slides in Hell
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *

Consider a metallic slide with 10 large holes in it equally spaced from
top to bottom.  If you attempt to slide down the slide you have a 50%
probability of sliding through each hole in the slide into an oleaginous
substance beneath the slide during each encounter with a hole.

1.  If you were a gambling person, which hole would you bet a person
would fall through?

2.  If you were a gambling person, how many attempts would it require
for a person to slide from the top of the slide to the bottom without
falling through a single hole.

3.  If all the people on earth lined up to go down the slide, and they
slid down a more horrifying slide with 100 holes at a rate of 1 person
per second, when would you expect the first person to arrive at the
bottom of the slide without falling through.
An hour? A day? A decade? ...
Received: from uoft02.utoledo.edu by watson.ibm.com (IBM VM SMTP V2R2) with TCP;
Title: Cliff Puzzle 12: Slides in Hell
>Consider a metallic slide with 10 large holes in it equally spaced from
>top to bottom.  If you attempt to slide down the slide you have a 50%
>probability of sliding through each hole in the slide into an
>oleaginous substance beneath the slide during each encounter with a
>hole.
>
>1.  If you were a gambling person, which hole would you bet a person
>would fall through?

None.  The best chance is the first hole but I got a 50-50 chance.  Why
bother?  (2nd hole is 1/4, 3rd 2**-3, ...)

>2.  If you were a gambling person, how many attempts would it require
>for a person to slide from the top of the slide to the bottom without
>falling through a single hole.

No gurantee.  Each slide is an independent event.  Now, if you are
talking mere probability, on the average, one in 1024 slides may make
it through all 10 holes.

>3.  If all the people on earth lined up to go down the slide, and they
>slid down a more horrifying slide with 100 holes at a rate of 1 person
>per second, when would you expect the first person to arrive at the
>bottom of the slide without falling through.  An hour? A day? A decade?

Again, can't tell.  It could be the first one, it could be none.  Probablity
can not foretell actual events.  But if you have infinite number of people
sliding down till eternity, on the average, you may see 1 person slide over
all holes every (2**100)/(365*24*69*6) years.  This number is many times
bigger than the world population for now.

==> pickover/pickover.12.s <==
-------------------------

In article <[e-mail]> you write:
: Consider a metallic slide with 10 large holes in it equally spaced from
: top to bottom.  If you attempt to slide down the slide you have a 50%
: probability of sliding through each hole in the slide into an oleaginous
: substance beneath the slide during each encounter with a hole.
:
: 1.  If you were a gambling person, which hole would you bet a person
: would fall through?
The chance of falling thru the first hole is 50%.  For the second hole, it
is (.5)(.5) = 25%, the thrid is (.5)^3 = .125.  The chance by the tenth
hole is about .0097 %.  Obviously, since I am limited to one hole, I would
place my money on hole #1 (best chance).

: 2.  If you were a gambling person, how many attempts would it require
: for a person to slide from the top of the slide to the bottom without
: falling through a single hole.
The sum of the prob for falling thru a hole is .5 + .5^2 + .5^3 +...+.5^10.
This is about 99.902% = .99902.  So about 98 times out of 100000, someone
will make it through without falling.  This is about 1 time out of 1020.
So give or take about 1020 tries....
:
: 3.  If all the people on earth lined up to go down the slide, and they
: slid down a more horrifying slide with 100 holes at a rate of 1 person
: per second, when would you expect the first person to arrive at the
: bottom of the slide without falling through.
: An hour? A day? A decade? ...
The prob for falling thru the last hole is .5^100 = 7.88x10^-31.  There must
be some chance less than this that one WILL make it thru the slide.  The MIN
number of tries that it must take is 1/.5^100 = 1.26x10^30.  At the given rate
this is about 9.647 x 10^23 years, much older than the universe if I remeber
correctly.
Also, the chance of making it must be GREATER than .5^101.  or with
all the math, the MAX amount of time is 1.929x10^24 years.  So give or
take about 1.5x10^24 years....



--
Michael Neylon   aka Masem the Great and Almighty Thermodynamics GOD!
      //         | Senior, Chemical Engineering, Univ. of Toledo
  \\ // Only the |   Summer Intern, NASA Lewis Research Center
\  \X/   AMIGA!  |         [e-mail]           /
 --------+ How do YOU spell 'potato'?  How 'bout 'lousy'? +----------
    "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L
-------------------------

In rec.puzzles you write:

>Title: Cliff Puzzle 12: Slides in Hell
>From: [e-mail]
>
>If you respond to this puzzle, if possible please send me your name,
>address, affiliation, e-mail address, so I can properly credit you if
Jeff Rogers
Rensselaer Polytechnic institute
[e-mail]

>Consider a metallic slide with 10 large holes in it equally spaced from
>top to bottom.  If you attempt to slide down the slide you have a 50%
>probability of sliding through each hole in the slide into an oleaginous
>substance beneath the slide during each encounter with a hole.
>
>1.  If you were a gambling person, which hole would you bet a person
>would fall through?

The first one. There's only a 50% chance of them getting past it, and a
small chance of them falling into each succeeding hole.
hole #     percent chance of reaching and falling into
1            50
2            25
3            12.5
4            6.25
5            3.125
6            1.5625
7            0.78125
8            0.390625
9            0.1953125
10           0.09765625

>
>2.  If you were a gambling person, how many attempts would it require
>for a person to slide from the top of the slide to the bottom without
>falling through a single hole.

The chances for reaching each succeeding hole are the same as reaching and
falling into the previous one. Therefore, the chances of passing all the
holes are the same as reaching and falling into the last hole  (see previous
answer for stats), which makes the probability .0009765625, so
statistically, 1024 slides would be required to guarantee reaching the
bottom. If I was a gambling person, I'd probably bet about half this,
because the actual events can happen in any order, and on average, I'd guess
that he'd get down in about 512 slides.

>
>3.  If all the people on earth lined up to go down the slide, and they
>slid down a more horrifying slide with 100 holes at a rate of 1 person
>per second, when would you expect the first person to arrive at the
>bottom of the slide without falling through.
>An hour? A day? A decade? ...

This is solved similarly; it is represented by powers of 2. To successfully
get past the last hole, it would require (statistically, at least) 2^100
or (by my trusty pocket calculator) 1.2676506 *10^30 slides.
More significant figures? dc! Which gives 1267650600228229401496703205376.
In similar logic as the last problem, I'd expect about half that, or
633825300114114700748351602688 slides. How much time would this be? Excluding
leap years, I calculate 20098468420665737593491 years. That's 20 sextillion
years, significantly more than the age of the universe, by about 11 orders
of magnitude. So I'd guess that no one will ever reach the bottom, they'll
all try and fail (assuming everyone only gets to go once), or die waiting in
line.

Diversion

--
"I can see 'em                          | "Want me to create a diversion?"
    I can see 'em                       | Diversion
        Someone wake me when it's over" | [e-mail]
-------------------------

In article <[e-mail]> you write:
Title: Cliff Puzzle 12: Slides in Hell
>Consider a metallic slide with 10 large holes in it equally spaced from
>top to bottom.  If you attempt to slide down the slide you have a 50%
>probability of sliding through each hole in the slide into an
>oleaginous substance beneath the slide during each encounter with a
>hole.
>
>1.  If you were a gambling person, which hole would you bet a person
>would fall through?

None.  The best chance is the first hole but I got a 50-50 chance.  Why
bother?  (2nd hole is 1/4, 3rd 2**-3, ...)

>2.  If you were a gambling person, how many attempts would it require
>for a person to slide from the top of the slide to the bottom without
>falling through a single hole.

No gurantee.  Each slide is an independent event.  Now, if you are
talking mere probability, on the average, one in 1024 slides may make
it through all 10 holes.

>3.  If all the people on earth lined up to go down the slide, and they
>slid down a more horrifying slide with 100 holes at a rate of 1 person
>per second, when would you expect the first person to arrive at the
>bottom of the slide without falling through.  An hour? A day? A decade?

Again, can't tell.  It could be the first one, it could be none.  Probablity
can not foretell actual events.  But if you have infinite number of people
sliding down till eternity, on the average, you may see 1 person slide over
all holes every (2**100)/(365*24*69*6) years.  This number is many times
bigger than the world population for now.
-------------------------

Some answers to your questions:

1. As the puzzle states there is a 50% chance of falling into each
hole, I would bet a person would fall into the first hole -- in a large
enough sample, 1/2 of the people will fall through the first hole, 1/4
through the second, 1/8 through the third, etc.

2. In a large sample, 1/(2^10) people would make it all the way down
the slide without falling through any of the holes (1/1024). This means
that 1023 out of 1024 people would fall through a hole. Using the
formula (1023/1024)^x=1/2, we can determine out of the first x people
to go down the slide, there is a 50% chance that one person will make
it down without falling through a hole.  The answer to this equation is
x=709.4 Thus I would bet that a person would make it all the way down
on one of the first 710 attempts.

3. As 2^100=1.2676*10^30 (roughly), and (including leaps years under
the Gregorian calendar) there are 31556952 seconds in the average year,
then statistically one person should make it down the slide every
4.017*10^22 YEARS. However, and this is a very rough estimate, I figure
the log of (1-1/(1.2676*10^30)) to be about -5.5*10^(-29). [I'm doing
the calculations on a scientific calculator which only has 10 places.]
Thus, using the formula xlog(1-1/2^100)=log(1/2), I get x=5.5*10^27.
Thus, there's about a 50% chance that after 5.5*10^27 seconds, someone
will have made it down the slide. To be on the safe side, I'd bet only
if I were given at least 6*10^27 seconds, a value which equals
1.901*10^20 YEARS.

I hope this answers the questions.

Ted Schuerzinger

email: [e-mail]
snailmail: HB 3819
Dartmouth College
Hanover, NH 03755
USA

In case you're wondering, I'm just a junior at Dartmouth who's
interested in puzzles like these. I'm not even a math major -- I'm a
double major in government and Russian.
-------------------------

In article <[e-mail]> you write:
>Title: Cliff Puzzle 12: Slides in Hell
>From: [e-mail]

>Consider a metallic slide with 10 large holes in it equally spaced from
>top to bottom.  If you attempt to slide down the slide you have a 50%
>probability of sliding through each hole in the slide into an oleaginous
>substance beneath the slide during each encounter with a hole.
>
>1.  If you were a gambling person, which hole would you bet a person
>would fall through?

There's a 50% chance of falling through the first hole, 25% the
second, 2^-n the n'th.  If the odds offered were the same, I'd go for
the first hole.

>2.  If you were a gambling person, how many attempts would it require
>for a person to slide from the top of the slide to the bottom without
>falling through a single hole.

You expect to make it 1 out of 1024 times; after 710 tries, the chance
of someone succeeding exceeds 1/2.  (Log base (1023/1024) of 1/2 is
709.4).

>3.  If all the people on earth lined up to go down the slide, and they
>slid down a more horrifying slide with 100 holes at a rate of 1 person
>per second, when would you expect the first person to arrive at the
>bottom of the slide without falling through.
>An hour? A day? A decade? ...

Never.  OK, 1/2^100 will make it.  There being under 2^33 people on
the planet, ...

After 4.2e22 years, the expected number of people who succeeded is 1;
after about 2.9e22 years, the chance of someone having succeeded is
about 1/2.

Like I said, never.

Seth        [e-mail]
-------------------------

In rec.puzzles you write:

>1.  If you were a gambling person, which hole would you bet a person
>would fall through?

If the pay-back odds were the same regardless of the hole, then obviously,
I'd bet on the first hole!  There's a 1:2 chance the person falls through
the first hole, a 1:4 combined chance of the person falling though the
second hole, etc...

>2.  If you were a gambling person, how many attempts would it require
>for a person to slide from the top of the slide to the bottom without
>falling through a single hole.

1024 is the median value for this case...  There's a 1:2**n chance of
a person falling through the nth hole, having missed all of the holes
before n.  Since the probability of falling through = the probability
passing over the hole safely (vs not ever getting there), the
probability that a person makes it to the end is also 1:1024.


>3.  If all the people on earth lined up to go down the slide, and they
>slid down a more horrifying slide with 100 holes at a rate of 1 person
>per second, when would you expect the first person to arrive at the
>bottom of the slide without falling through.
>An hour? A day? A decade? ...
There is a 1:2**(100-Log2(5 billion people)) chance that somebody makes
it through...  Given a finite # of people on the planet (approx 5 bil.)
I think we'll run out first...


--Joseph Zbiciak   [e-mail]


-------------------------

Subject: Re: Cliff Puzzle 12: Slides in Hell (SPOILER)
Newsgroups: rec.puzzles
References: <[e-mail]>

In article <[e-mail]>, Cliff Pickover writes:

> Consider a metallic slide with 10 large holes in it equally spaced from
> top to bottom.  If you attempt to slide down the slide you have a 50%
> probability of sliding through each hole in the slide into an oleaginous
> substance beneath the slide during each encounter with a hole.

> 1.  If you were a gambling person, which hole would you bet a person
> would fall through?

The probability of falling into hole i is (1/2)^i, so your best bet
would be hole 1.

> 2.  If you were a gambling person, how many attempts would it require
> for a person to slide from the top of the slide to the bottom without
> falling through a single hole.

The probability of success is p = (1/2)^10, and as each trial is
independant the expected number of trials before success is 1/p or
2^10.

> 3.  If all the people on earth lined up to go down the slide, and they
> slid down a more horrifying slide with 100 holes at a rate of 1 person
> per second, when would you expect the first person to arrive at the
> bottom of the slide without falling through.

In this case the number of expected trials is 2^100, which is much
larger than the total number of people.

> An hour? A day? A decade? ...

Try about 10^24 years.  As another problem, assuming a large enough
supply of sliders estimate when the slide will wear through from
friction.
-------------------------

In article <[e-mail]> you write:
>Title: Cliff Puzzle 12: Slides in Hell
>From: [e-mail]
>
>If you respond to this puzzle, if possible please send me your name,
>address, affiliation, e-mail address, so I can properly credit you if
>you provide unique information.  PLEASE ALSO directly mail me a copy of
>your response in addition to any responding you do in the newsgroup.  I
>will assume it is OK to describe your answer in any article or
>publication I may write in the future, with attribution to you, unless
>you state otherwise.  Thanks, Cliff Pickover
>
>      * * *
>
>Consider a metallic slide with 10 large holes in it equally spaced from
>top to bottom.  If you attempt to slide down the slide you have a 50%
>probability of sliding through each hole in the slide into an oleaginous
>substance beneath the slide during each encounter with a hole.
>
>1.  If you were a gambling person, which hole would you bet a person
>would fall through?

I'd bet that they fell through the first hole.  The probability of that
happening is 50%.  The probability of them falling through the second
hole is:
P(didn't fall through the first)*P(fell through the second) = 50%*50% = 25%

In general, P(falls through hole n)=
P(no fall through 1)*P(no fall through 2)*...*P(no fall through n-1)
 *P(fell through hole n).
For this problem, P(falls through hole n) is (50%)^n, where n is the hole #
from the top.

>2.  If you were a gambling person, how many attempts would it require
>for a person to slide from the top of the slide to the bottom without
>falling through a single hole.

(Hey, after the first failed attempt, they're screwed, no?)
P(success)=P(no fail)=P(no fall 1)P(no fall 2)...P(no fall 10)
 =50%^10
 =1/1024
They should make it at least one time in 1024.

>3.  If all the people on earth lined up to go down the slide, and they
>slid down a more horrifying slide with 100 holes at a rate of 1 person
>per second, when would you expect the first person to arrive at the
>bottom of the slide without falling through.
>An hour? A day? A decade? ...

Oh, one in about 4.02*10^22 years...  I wouldn't hold my breath.


-Richard
-------------------------

1. I would bet on the first hole, as there is a 0.5 probability of a person's
   falling into it, which is the highest such probability.

2. The probability of reaching the end of the slide on a particular try is
   1/2^10 = 1/1024.  In 709 tries, there is an approximately 0.5 probability of

3. Beats me - the even money bet is for a number of tries (approximately) equal
          ((2^100 - 1)/(2^100))
   calculate it.



--
_______________________________________________________________________
Dan Blum           Institute for the Learning Sciences   Room 327
[e-mail]   1890 Maple Ave., Evanston, IL 60201   708-467-2306

"Let it be granted that a controversy may be raised about any question,
 and at any distance from that question."
                      Lewis Carroll
_______________________________________________________________________


==> pickover/pickover.13.p <==
Title: Cliff Puzzle 13: Ladders to Heaven
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *

Consider the following scenario.  A standard ladder stretches from each
country on the earth upward a distance equal to the distance from the
earth to the moon.

Assume:
1. the ladder is made out of a strong metal such as
titanium, which will not break.
2. the ladder is inclined at a very steep angle, 70 degrees, for
each country.
3. there is a breathable atmosphere.
4. the people (or teams of people) are allowed to use standard
mountain climbing and camping gear, e.g. ropes, backpacks, etc. but not
sophisticated electrical mechanisms, engines, etc.
5. a reward is given to whomever reaches the top of the ladder
first: 1 million dollars to that person.  In addition the country's
national debt is wiped out.

Questions:
1.  Approximate how long it would take a person (or team of people) to
reach the top of the ladder.  Days?  Weeks?  Years?

2. Which country would be the first?

3. Is there any novel method you would suggest to achieve this goal?

4. Is this task impossible to carry out.

==> pickover/pickover.13.s <==
-------------------------

Interesting puzzle... Just one question though: Is there a moon,
i.e. is it possible to use the gravitational field of the moon to your
advantage by "falling upwards" once you have reached the point where
the moon's gravity is bigger than the erath's (and do we also assume that
the the climber(s) must survive the fall?? :-) or shall we assume that the
earth is alone in the universe?


Spyros Potamianos
[e-mail]
-------------------------

Newsgroups: rec.puzzles
Subject: Re: Cliff Puzzle 13: Ladders to Heaven
References: <[e-mail]>
Organization: The Chrome Plated Megaphone of Destiny

>1.  Approximate how long it would take a person (or team of people) to
>reach the top of the ladder.  Days?  Weeks?  Years?

Note that after you're 22,300 miles from the earth's axis, you get to
"fall" the rest of the way, as long as you don't lose contact with
the ladder.

>2. Which country would be the first?

It has already been pointed out that countries on the equator have an
advantage.  I suppose you could consider that countries with a large
national debt have extra motivation.  :-)

>3. Is there any novel method you would suggest to achieve this goal?

I would suggest a bicycle-like vehicle clamped to the ladder.  By
pulling a light but strong rope on a pulley (perhaps obtained form
the same source as this fantastic ladder material), riders could be
changed fairly quickly, thanks to a crew of brawny pulley-pullers
with a variable-geared linkage to the rope.

For the rider to pull this ever-longer rope seems impossible, but I
think shorter segments could be lifted and linked.  Or the ground
crew could help the rider by pulling down rope from a hub of lesser
diameter than the wheels of the vehicle.

>4. Is this task impossible to carry out.

No.  I thought it might be impossible to halt at the far end of the
ladder and return, due to centrifugal acceleration, but that
acceleration turns out to be only about 5 cm/s^2.
__________________________________________________________
Matt Crawford       [e-mail]       Java Man


-------------------------

> How do we get food to the people?

I would have the riders change so often that they'd only need some
high-carbohydrate snacks and a couple quarts of fluid.  I think the
brawny ground crew could pull up the next rider, with his supplies
and another pulley and segment of rope, at an acceleration of about
0.5 g or better.  That would be under 90 minutes for each shift-
change up to the synchronous orbit level.

I haven't figured out yet how to link each new piece of rope that's
pulled up with a rider to the pulley that's at the high point reached
by the previous rider.  Linking is easy, but it would be nice to find
a way that lets the next pulled-up rider go from one segment to the
other without interruption.  Well, since the sky-buckets at
Disneyland do this trick at each end, I know it can be done.

I didn't know you'd written any books, but it was clear you're
working on one now.  Sure, send a list, but I have access to some
on-line catalogs, so maybe I can find them anyway.

            Matt Crawford
-------------------------

> Consider the following scenario.  A standard ladder stretches from each
> country on the earth upward a distance equal to the distance from the
> earth to the moon.
>
> Assume:
> 1. the ladder is made out of a strong metal such as
> titanium, which will not break.
> 2. the ladder is inclined at a very steep angle, 70 degrees, for
> each country.
> 3. there is a breathable atmosphere.
> 4. the people (or teams of people) are allowed to use standard
> mountain climbing and camping gear, e.g. ropes, backpacks, etc. but not
> sophisticated electrical mechanisms, engines, etc.
> 5. a reward is given to whomever reaches the top of the ladder
> first: 1 million dollars to that person.  In addition the country's
> national debt is wiped out.

I would imagine that one would be able to fashion a hot air balloon given
condition 4.  Also, given condition 3, the hot air balloon would be able
to cover the entire distance.  One would then only need to attach a sliding
hookup between the ladder and the balloon and wait.

===M.Graf==[e-mail]==================================================


==> pickover/pickover.14.p <==
Title: Cliff Puzzle 14: Geography Genuflection
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *

1.  How would the world be different today, geopolitically speaking, if
the ancient land masses had never drifted apart and, therefore,
today's world consisted of a single supercontintent?

2.  What would today's world be like if the land mass which formed the
Greek peninsula never existed?

3.  What would today's world be like if the land bridge which joined
Alaska to Asia never existed?

4.  Why do all the major peninsulas on earth point south?  See for
example:  Italy, Greece, Florida, and Baja, and the tips of Africa,
South America, India, Norway, Sweden, Greenland, and many other
landmasses.

==> pickover/pickover.14.s <==
-------------------------

In rec.puzzles you write:

>If you respond to this puzzle, if possible please send me your name,
>address, affiliation, e-mail address, so I can properly credit you if
>you provide unique information.
>
Mike Neergaard
University of Wisconsin
[e-mail]

I'm not a professional at this sort of thing, so I just summarized my
conclusions.  I'm sure they would be ripped to shreds by any competent
whatsit-type-individual-who-knows-all-about-this-kind-of-stuff.

>1.  How would the world be different today, geopolitically speaking, if
>the ancient land masses had never drifted apart and, therefore,
>today's world consisted of a single supercontintent?
We would all speak German.

>2.  What would today's world be like if the land mass which formed the
>Greek peninsula never existed?
>
We would know a low more about fluid dynamics.

>3.  What would today's world be like if the land bridge which joined
>Alaska to Asia never existed?
Christopher Columbus would be a national hero, instead of being vulnerable
to counter-claims of genocide.  America would have been settled several
decades later, due to a dearth of demonstrable natural resources.

>4.  Why do all the major peninsulas on earth point south?  See for
>example:  Italy, Greece, Florida, and Baja, and the tips of Africa,
>South America, India, Norway, Sweden, Greenland, and many other
>landmasses.
I just work here . . .
--
I really don't make any claim at all to know what I'm talking about.
Actually, I make no claim to know what YOU'RE talking about, either.
In fact, now I've forgotten what we were talking about . . .

-------------------------

In article <[e-mail]> you write:
>Title: Cliff Puzzle 14: Geography Genuflection
>From: [e-mail]
>
>If you respond to this puzzle, if possible please send me your name,
>address, affiliation, e-mail address, so I can properly credit you if
>you provide unique information.  PLEASE ALSO directly mail me a copy of
>your response in addition to any responding you do in the newsgroup.  I
>will assume it is OK to describe your answer in any article or
>publication I may write in the future, with attribution to you, unless
>you state otherwise.  Thanks, Cliff Pickover
>
>      * * *
>

Okay, administrative trivia first.  My name is Martin Eiger, you don't
need my address (home or business?), I don't want you citing my
affiliation if you quote me, and my e-mail address is
[e-mail].

>1.  How would the world be different today, geopolitically speaking, if
>the ancient land masses had never drifted apart and, therefore,
>today's world consisted of a single supercontintent?

My theory is that mankind would never have evolved.  The dominant
species would still be some sort of mammal, but not us.  This renders
a large number of geopolitical questions irrelevant.  For example,
elephant-like creatures are unlikely to care whether there is one or
two Germanys.


>2.  What would today's world be like if the land mass which formed the
>Greek peninsula never existed?

A tough one, since I'm not up on my Greek influences in the evolution
of civilization.  My guess is that civilization would have evolved
anyway, probably not too differently than it did.  It might not have
evolved as fast, i.e., we might now be where we were a thousand years
ago or so, but over the long haul, human history would follow a
similar course.


>3.  What would today's world be like if the land bridge which joined
>Alaska to Asia never existed?

Pretty much the same, I bet.  People would have colonized North
America anyway.  After all, they got to Hawaii, so somebody could
probably have gotten to North America.  And whether or not people
colonized North America from across the Pacific, people from Europe
would have paved the place over just the same.


>4.  Why do all the major peninsulas on earth point south?  See for
>example:  Italy, Greece, Florida, and Baja, and the tips of Africa,
>South America, India, Norway, Sweden, Greenland, and many other
>landmasses.

First of all, you have to define what's a major peninsula.  Secondly,
I don't like your list.  Norway and Sweden are on the same peninsula,
and Greenland is an island, not a peninsula.  And third, there are
plenty of perfectly fine peninsulas that don't point south:  Alaska,
Siberia, Michigan (two peninsulas for the price of one), Yucatan,
Arabia (points kind of southeast), and Iberia, for instance.  And
fourth, you missed a few good southern-pointing ones, such as Korea,
Crimea, the Sinai, and the one that kind of points from eastern
Siberia toward Japan that I'm sure has a name but I don't know it.  So
while there are lots of peninsulas pointing lots of directions, a
majority of them do seem to point south, and I have no idea why.

==> pickover/pickover.15.p <==
Title: Cliff Puzzle 15: Cherries in Wine Glasses
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *

Consider a 9x9 grid of beautiful crystal wineglasses.  Throw 32 cherries
at the grid.  A glass is considered occupied if it contains at least one
cherry.  (With each throw a cherry goes into one of the glasses.)  How
many different patterns of occupied glasses can you make?  (A glass with
more than one cherry is considered the same as a glass with one cherry
in the pattern).

2.  Same as above except that you place 8 cherries in glasses (x,y) and
then determine the other positions by placing cherries at (x,-y),
(-x,y), (-x,-y) leading to 32 cherries in the grid.  Consider the array
of glasses centered at the origin.  How many different patterns of
occupied glasses can you make?  (A glass with more than one cherry is
considered the same as a glass with one cherry in the pattern).

3. Can your results be extrapolated to an NxN grid with M cherries
thrown at it for both problems?


==> pickover/pickover.15.s <==
In article <[e-mail]> you write:
: Consider a 9x9 grid of beautiful crystal wineglasses.  Throw 32 cherries
: at the grid.  A glass is considered occupied if it contains at least one
: cherry.  (With each throw a cherry goes into one of the glasses.)  How
: many different patterns of occupied glasses can you make?  (A glass with
: more than one cherry is considered the same as a glass with one cherry
: in the pattern).
Assuming that rotated patterns are allowed, then it is (simply)
sum( 81!/(81-n)! , n=1->32) . Since, if a total of n different classes are
filled, then the number of combinations is 81!/(81-n)!.  Since there can
be from 1 to 32 glasses filled, the total # is just the sum of these...

:
: 2.  Same as above except that you place 8 cherries in glasses (x,y) and
: then determine the other positions by placing cherries at (x,-y),
: (-x,y), (-x,-y) leading to 32 cherries in the grid.  Consider the array
: of glasses centered at the origin.  How many different patterns of
: occupied glasses can you make?  (A glass with more than one cherry is
: considered the same as a glass with one cherry in the pattern).
This limitation basically reduces the number of available spots, from 9x9
to 5x5.  Also, I only have to worry about 8 occupied spaces.  Soo...
#of comb. = sum( (25!/(25-n)!, n=1->8)
:
: 3. Can your results be extrapolated to an NxN grid with M cherries
: thrown at it for both problems?
With a odd N, and M = 4k (evenly divs by 4), then
for 1....
#of comb = sum( (N^2)!/(N^2-n)!  , n=1->M)
for 2....
#of comb = sum( (((N+1)/2)^2)!/(((N+1)/2)^2-n)! , n=1->M/4)

--
Michael Neylon   aka Masem the Great and Almighty Thermodynamics GOD!
      //         | Senior, Chemical Engineering, Univ. of Toledo
  \\ // Only the |   Summer Intern, NASA Lewis Research Center
\  \X/   AMIGA!  |         [e-mail]           /
 --------+ How do YOU spell 'potato'?  How 'bout 'lousy'? +----------
    "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L

==> pickover/pickover.16.p <==
Title: Cliff Puzzle 16: Undulating Squares
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *

A square number is of the form y=x**2.  For example, 25 is a square
number.

Undulating numbers are of the form:  ababababab... For example, the
following are undulating numbers:  1717171, 282828, etc.

1. Are there any undulating square numbers?

2. Are there any undulating cube numbers?


==> pickover/pickover.16.s <==
-------------------------

In article <[e-mail]> you write:
: 1. Are there any undulating square numbers?
11^2 = 121

: 2. Are there any undulating cube numbers?
7^3 = 343

(yes, I know they're short, but they qualify!)

--
Michael Neylon   aka Masem the Great and Almighty Thermodynamics GOD!
      //         | Senior, Chemical Engineering, Univ. of Toledo
  \\ // Only the |   Summer Intern, NASA Lewis Research Center
\  \X/   AMIGA!  |         [e-mail]           /
 --------+ How do YOU spell 'potato'?  How 'bout 'lousy'? +----------
    "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L
-------------------------

In article <[e-mail]> you write:
>Hi, I was interested in non-trivial cases.  Those with greater
>than 3 digits.  Award goes to the person who finds the largest
>undulating square or cube number.  Thanks, Cliff

343 and 676 aren't trivial (unlike 121 and 484 it doesn't come from
obvious algebraic identities).  The chance that a "random"
number around x should be a perfect square is about 1/sqrt(x);
more generally, x^(-1+1/d) for a perfect d-th power.  Since
there are for each k only 90 k-digit undulants you expect
to find only finitely many of these that are perfect powers,
and none that are very large.  But provably listing all cases
is probably only barely, if at all, possible by present-day
methods for treating exponential Diophantine equations, unless
(as was shown in a rec.puzzles posting re your puzzles on
arith. prog. of squares with common difference 10^k) there is
some ad-hoc trick available.  At any rate the largest undulating
power is probably 69696=264^2, though 211^3=9393931 comes
remarkably close.

--Noam D. Elkies
-------------------------

In article <[e-mail]>, you write...
>1. Are there any undulating square numbers?
>
    Other than the obvious 11**2, 22**2, and 26**2, there is 264**2
which equals 69696.

>2. Are there any undulating cube numbers?
>
    Just 7**3 as far as I can tell, though I'm limited to IEEE computational
reals.

PauL M SchwartZ  (-Z-)  |  Follow men's eyes as they look to the skies
[e-mail]   |         the shifting shafts of shining
[e-mail]    |        weave the fabric of their dreams
[e-mail]    |                    - RUSH -


==> pickover/pickover.17.p <==
Title: Cliff Puzzle 17: Weird Recursive Sequence
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *


Consider the simple yet weird recursive formula

a(n) = a(a(n-1)) + a(n-a(n-1))

The sequences starts with a(1) = 1, and a(2) = 1.  The "future" values
at higher values of n depend on past values in intricate recursive ways.
Can you determined the third member of the sequence?  At first, this may
seem a little complicated to evaluate, but you can being slowly, by
inserting values for n, as in the following:

a(3) = a(a(2)) + a(3-a(2))
a(3) = a(1) + a(3-1) =
a(3) = 1+1 = 2

Therefore, the 3rd value of the sequence a(3) is 2.

The sequence a(n) seems simple enough: 1, 1, 2, 2, 3, 4, 4, 4, 5, ...
Try computing a few additional numbers.  Can you find any interesting
patterns?  The prolific mathematician John H Conway presented this
recursive sequence at a recent talk entitled "Some Crazy Sequences."  He
noticed that the value a(n)/n approaches 1/2 as the sequence grows and n
becomes larger.  Can you find a value, N, above which the sequence the
value of a(n)/n is always within 0.05 of the value 1/2?  (In other
words,
.eq vbar a(n)/n -1/2 vbar lt 0.05.
The bars indicate the absolute value.)

   A difficult problem? you ask.
John Conway offered $10,000 to the person to find the s-m-a-l-l-e-s-t
such N. A month after Conway made the offer, Colin Mallows of AT&T
solved the $10,000 question:  N = 3,173,375,556.  Manfred Shroeder has
noted that the sequence is "replete with appealing self-similarities
that contain the clue to the problem's solution."  Can you find any
self-similarities?  As I write this, no-one on the planet has found a
value for the smallest N such that a(n)/n is always within 0.01 of the
value 1/2.
.eq (vbar a(n)/n -1/2 vbar lt 0.01. )



==> pickover/pickover.17.s <==
-------------------------

In article <[e-mail]> you write:
: Title: Cliff Puzzle 17: Weird Recursive Sequence
: Consider the simple yet weird recursive formula
: a(n) = a(a(n-1)) + a(n-a(n-1))

The first 32 terms, and the ratio a(n)/n for each is as follows...

n   a(n)     a(n)/n
1   1        1.0
2   1        1.0
3   2        .666
4   2        .5
5   3        .6
6   4        .666
7   4        .5714
8   4        .5
9   5        .5555
10  6        .6
11  7        .6363
12  7        .5833
13  8        .6153
14  8        .5714
15  8        .5333
16  8        .5
17  9        .5294
18  10       .5555
19  11       .5789
20  12       .6
21  12       .5714
22  13       .5909
23  14       .6086
24  14       .5833
25  15       .6
26  15       .5769
27  15       .5555
28  16       .5714
29  16       .5517
30  16       .5333
31  16       .5161
32  16       .5
33  17 .... and so and....

off the top, we can see that on the 2^k (k a pos. int) terms, the
ratio goes to .5

between each of these, the ratio goes up and then drops back to .5
(ignoring the variances due to integer arithmatic)

the value of n at the maximum in each jump is halfway between the two
2^k points.  The value of a(n) at those points seems to be
2^(k-1) - f(k), where f(k) is some function that I cannot determine
without more computing power.... *sniff*

Therefore, we must find a value of x such that...
(2^(x-1)-f(x))/2^x - .5 <.05 (or whatever)
or
f(x)/2^x < .05

and then N would be .5*(2^x-2^(x-1))

if I could see the next terms up to 128, I might be able to calculate it...


--
Michael Neylon   aka Masem the Great and Almighty Thermodynamics GOD!
      //         | Senior, Chemical Engineering, Univ. of Toledo
  \\ // Only the |   Summer Intern, NASA Lewis Research Center
\  \X/   AMIGA!  |         [e-mail]           /
 --------+ How do YOU spell 'potato'?  How 'bout 'lousy'? +----------
    "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L
-------------------------

In article <[e-mail]> you write:

>John Conway offered $10,000 to the person to find the s-m-a-l-l-e-s-t
>such N. A month after Conway made the offer, Colin Mallows of AT&T
>solved the $10,000 question:  N = 3,173,375,556.

As I pointed out in my posting, this is incorrect, and differs from
Mallows' correct answer published in his article. But a bit of
investigation shows that the above N is hardly a random guess, either.
Conway's sequence is best understood by analyzing it on "levels",
where the   k'th level is the set of integers between  2^k  and 2^(k+1).
It turns out that Mallows' correct answer, 6083008742, lies on level 32,
and the largest candidate answer on level 31 is  N=3173375556, the
number quoted above.

Where did you see the above value of N given as the answer to Conway's
question?

-tal    [e-mail]

p.s.  As I found out when I edited my posted response to your message,
      you either use lines longer than 80 characters in your postings,
      or else your editor appends extra linefeeds to each line.  Since
      both conditions could be problematic for a lot of people who read
      your messages on rec.puzzles, you might want to correct this
      condition.


==> pickover/pickover.18.p <==
Title: Cliff Puzzle 18: Difficult Nested Roots
From: [e-mail]

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE ALSO directly mail me a copy of
your response in addition to any responding you do in the newsgroup.  I
will assume it is OK to describe your answer in any article or
publication I may write in the future, with attribution to you, unless
you state otherwise.  Thanks, Cliff Pickover

      * * *

Consider the following nested set of square roots.

.eq ? = sqrt <1 + G sqrt <1+(G+1) sqrt < 1 + ... >>>

Here, G indicates "Googol" or  10**100.
The "<" and ">" symbols indicate where the beginning and ends of the
the nested roots.

1. What is the value for in this infinite set of nested roots.
2. What is the next term under the root?

Hint:
In 1911, the famous mathematical prodigy Srinivasa Ramanujan posed the
following question (#298) in a new mathematical journal called the
:Journal of the Indian Mathematical Society.

.eq ? = sqrt <1 + 2 sqrt <1+3 sqrt <1 + ... >>>


==> pickover/pickover.18.s <==
-------------------------

In article <[e-mail]> you write:
: Title: Cliff Puzzle 18: Difficult Nested Roots
: From: [e-mail]
: Consider the following nested set of square roots.
:
:  ? = sqrt <1 + G sqrt <1+(G+1) sqrt < 1 + ... >>>
:
: Here, G indicates "Googol" or  10**100.
: The "<" and ">" symbols indicate where the beginning and ends of the
: the nested roots.
:
: 1. What is the value for in this infinite set of nested roots.
: 2. What is the next term under the root?
: Hint:
: In 1911, a twenty-three-year-old Indian clerk named Srinivasa Ramanujan
: posed the following question (#298) in a new mathematical journal called
: the Journal of the Indian Mathematical Society.
:
:  ? = sqrt <1 + 2 sqrt <1+3 sqrt <1 + ... >>>
:
Doing a n-depth thing-a-ding on this.....
n=1   v=1
2     1.732
3     2.236
4     2.5598
5     2.7551
6     2.867
....
20    2.99999376
....
so I expect that the sum is actually 3.  Or in the general case when the
2 (or the G from above) is replaced by m, then the evaluation of the series
is m+1.  This CAN be shown as follows....

m+1 = sqrt(1+m sqrt(1+(m+1)*sqrt(....))
m^2 + 2m +1 = 1 + m *sqrt(1 + (m+1)*sqrt(...))
m^2 + 2m = m*sqrt(1+(m+1)*sqrt(...))
m+2 = sqrt(1+(m+1)*sqrt(1+(m+2)*sqrt(...))

Thus if m+1 is then sum when the series is based off m, then m+2 is then
sum when the series is based off m+1.  Since this works for m=2 (as shown
above), then it must work for all whole numbers (mathematical induction is
such a wonderful thing...)

Therefore, the sum with m=G is G+1.

The next term, as show above, is (1+(m+2)*sqrt(1+....))


--
Michael Neylon   aka Masem the Great and Almighty Thermodynamics GOD!
      //         | Senior, Chemical Engineering, Univ. of Toledo
  \\ // Only the |   Summer Intern, NASA Lewis Research Center
\  \X/   AMIGA!  |         [e-mail]           /
 --------+ How do YOU spell 'potato'?  How 'bout 'lousy'? +----------
    "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L