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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

378 pickover/pickover.02.p


Title: Cliff Puzzle 2: Grid of the Gods
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name,
address, affiliation, e-mail address. If you like, tell me a little bit
about yourself. You might 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 grid of infinitesimal dots spaced 1 inch apart in a cube with
an edge equal in length to the diameter of the sun (4.5x10**9 feet).
For conceptual purposes, you can think of the dots as having unit
spacing, being precisely placed at 1.00000...., 2.00000....,
3.00000...., etc. Next choose one of the dots and draw a line through it
which extends from that dot to the edge of the huge cube in both
directions.

Stop And Think

1. What is the probability that your line will intersect another dot
in the fine grid of dots within the cube the size of the sun?
Would your answer be different if the cube were the size of the
solar system?

2. Could a computer program be written to simulate this process?

3. Answer the two questions above, but this time assume the line
to have some finite thickness, T.

pickover/pickover.02.s

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

In article <1992Sep14.141551.42075@watson.ibm.com> you write:
>Title: Cliff Puzzle 2: Grid of the Gods
>From: cliff@watson.ibm.com
>
>If you respond to this puzzle, if possible please include your name,
>address, affiliation, e-mail address. If you like, tell me a little bit
>about yourself. You might 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 grid of infinitesimal dots spaced 1 inch apart in a cube with
>an edge equal in length to the diameter of the sun (4.5x10**9 feet).
>For conceptual purposes, you can think of the dots as having unit
>spacing, being precisely placed at 1.00000...., 2.00000....,
>3.00000...., etc. Next choose one of the dots and draw a line through it
>which extends from that dot to the edge of the huge cube in both
>directions.
>
>Stop And Think
>
>1. What is the probability that your line will intersect another dot
>in the fine grid of dots within the cube the size of the sun?
>Would your answer be different if the cube were the size of the
>solar system?

That depends on the manner the dot and the direction of the line were choosen.
If both process used uniform (or any other continous) distribution, then - of
course - the probability would be zero. If, for instance, the direction of
the line is always choosen to be parallel to one of the cube's edges, then the
probability is one.

>
>2. Could a computer program be written to simulate this process?

Not a meaningfull question. Simple minded programs could never simulate
infinitesimal points, but well thought out algorithm could express anything
that can be shown analytically.
>
>3. Answer the two questions above, but this time assume the line
>to have some finite thickness, T.
>

This is equivelent to making each dot of diameter T, and keeping the line thin.
For T> (1 inch / 4.5*10^9 ft) inches, the probability -> 1.

A simple minded computer program could simulate this.

Dan Shoham
shoham@ll.mit.edu
-------------------------

In article <1992Sep14.141551.42075@watson.ibm.com> you write:
>1. What is the probability that your line will intersect another dot
>in the fine grid of dots within the cube the size of the sun?

About 50%, because I usually draw horizontal lines.

I.e., YOU DIDN'T GIVE THE DISTRIBUTION OF "lines".

cf the puzzle of "what is the probability that a randomly selected
chord of a circle is longer than the radius of that circle." The
answer depends on how you "randomly select."
_________________________________________________________
Matt Crawford crawdad@fncent.fnal.gov Fermilab

 

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