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

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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
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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