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

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Alan Sagan

Posts: 13
Registered: 12/13/04
Re: Billiards Puzzle
Posted: Oct 1, 2004 3:17 PM
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Please ignore my last post since it was in error, sorry
Puzzle used as spoiler space (Acid Pooh) wrote in message news:<>...
> Here's a neat little puzzle I thought of, though I still haven't
> figured out its answer.
> Suppose you're racking up 15 billiard balls in one of the standard
> configurations (I'm not going to try to typeset these, so just picture
> an equilateral triangle instead of a right one):
> S
> T S
> S E T
> T S T S
> S T S T T
> where S is a "solid," T is a stripe, and E is the eight ball. A
> configuration is also standard if every S is mapped to a T, or if the
> triangle is reflected across its verticle axis of symmetry. Anyway,
> so you're racking up and you dump 15 balls into the rack randomly.
> Assuming you don't make any mistakes, what's the maximum number of two
> ball permuations necessary to get to any of the 4 standard
> configurations?
> 'cid 'ooh

takes 4 moves

I think it easy too show that 4 is the max.
Leaving the eight ball in place take any configuration of S&T. Now
compare this to a specific standard configuration, call it ConfigX.
Suppose you find N S-balls are out of place where N=0-7. If you swap S
and T in ConfigX than 7-N S-balls will now be out of place. So I can
always pick a configuration where the maximum number of S-balls that
can be out of place is 3 so we have a total of 6 balls out of place.
Now swap out the eight ball and you have 8 balls out of place, which
is 4 swaps. Done.

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