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Topic: Billiards Puzzle
Replies: 8   Last Post: Oct 3, 2004 4:23 AM

 Messages: [ Previous | Next ]
 Glenn C. Rhoads Posts: 31 Registered: 12/13/04
Re: Billiards Puzzle
Posted: Oct 3, 2004 1:42 AM

Michael Mendelsohn &lt;keine.Werbung.1300@msgid.michael.mendelsohn.de&gt; wrote in message news:&lt;415D354A.A23E7197@msgid.michael.mendelsohn.de&gt;...
&gt;&gt; poopdeville@gmail.com (Acid Pooh) wrote in
&gt;&gt;&gt; Suppose you're racking up 15 billiard balls in one of the standard
&gt;&gt;&gt; configurations (I'm not going to try to typeset these, so just picture
&gt;&gt;&gt; an equilateral triangle instead of a right one):
&gt;&gt;&gt;
&gt;&gt;&gt; S
&gt;&gt;&gt; T S
&gt;&gt;&gt; S E T
&gt;&gt;&gt; T S T S
&gt;&gt;&gt; S T S T T
&gt;&gt;&gt;
&gt;&gt;&gt; where S is a "solid," T is a stripe, and E is the eight ball. A
&gt;&gt;&gt; configuration is also standard if every S is mapped to a T, or if the
&gt;&gt;&gt; triangle is reflected across its verticle axis of symmetry. Anyway,
&gt;&gt;&gt; so you're racking up and you dump 15 balls into the rack randomly.
&gt;&gt;&gt; Assuming you don't make any mistakes, what's the maximum number of two
&gt;&gt;&gt; ball permuations necessary to get to any of the 4 standard
&gt;&gt;&gt; configurations?
&gt;&gt;
&gt;&gt; I don't understand your description of "standard configuration."
&gt;&gt; What do you mean by "every S is mapped to a T"?
&gt;
&gt; I understand this to mean that
&gt;
&gt; S
&gt; T S
&gt; S E T
&gt; T S T S
&gt; S T S T T
&gt;
&gt; maps to
&gt;
&gt; T
&gt; S T
&gt; T E S
&gt; S T S T
&gt; T S T S S

But you can perform such a mapping on *any* rack. Why is the
above configuration standard and some other configuration not?
What constraints does a standard configuration have to satisfy?

The post seems to be defining a configuration as standard
if you can transform it standard configuration. This is a
circular and meaningless definition.

&gt;&gt; Also, the two back corners cannot be the same (by the
&gt;&gt; rules of the 8 ball) and hence, the rack is never symmetric across
&gt;&gt; the vertical axis.
&gt;
&gt; Well, though the rack is not symmetric, the triangle is; which means
&gt; that the above setups reflect to those setups below:
&gt;
&gt;
&gt; S
&gt; S T
&gt; T E S
&gt; S T S T
&gt; T T S T S
&gt;
&gt; T
&gt; T S
&gt; S E T
&gt; T S T S
&gt; S S T S T

Ditto.

&gt; These appear to be the only 4 legal billiard setups.

Now that is a clear definition -- these are the four standard
configurations. In retrospect, it seems the original poster
was trying to avoid simply listing these four configurations
by saying, the standard configurations are this configuration
plus the configurations that can be obtained from it by
performing any combination of these two transformations.
That's not the way I read it but the description is sufficient
for the purposes of defining the puzzle.

&gt; (Although I can't
&gt; fathom why the bottom row couldn't be T S S T S, for example.)

I can't help you because I do not know all of the constraints
that a standard rack in 8 ball is supposed to satisfy.

&gt; The problem then boils down to what the maximum of the minimums of
&gt; 2-ball permutations from any permutation to any of these 4 permutations
&gt; is. ;)
&gt;
&gt; Cheers
&gt; Michael

Date Subject Author
9/30/04 Acid Pooh
10/1/04 Alan Sagan
10/1/04 Michael Mendelsohn
10/1/04 Michael Mendelsohn