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        riddles >> easy >> Billiard Ball Problem
        
(Message started by: cool_joh on Nov 21^st, 2007, 10:19pm)

Title: Billiard Ball Problem
Post by cool_joh on Nov 21^st, 2007, 10:19pm

Suppose we place a ball on a billiard table at some arbitrary position. We wish to drive the ball in such a way that it will bounce off the four sides of the table and the pass through its original position. What direction should we drive the ball?

Title: Re: Billiard Ball Problem
Post by towr on Nov 21^st, 2007, 11:25pm

It seems, [hide]parallel to the diagonal[/hide]

Title: Re: Billiard Ball Problem
Post by cool_joh on Nov 22^nd, 2007, 1:03am

on 11/21/07 at 23:25:39, towr wrote:

It seems, [hide]parallel to the diagonal[/hide]

How can you prove it?

Title: Re: Billiard Ball Problem
Post by towr on Nov 22^nd, 2007, 4:36am

[hide]Use reflections to expand the single table to 9 in a 3*3 grid, then aim the ball from it's place in one corner-table of the arrangement to the same place in th eoppozite corner-table; the ball will move virtually two table-length down (or up) and two table-withs to the left (or right), so it's direction is parallel to the table diagonal.
And necessarily it crosses 4 different borders, so there's 4 bounces.[/hide]

Title: Re: Billiard Ball Problem
Post by Grimbal on Nov 22^nd, 2007, 5:08am

[hide] It is also necessary.
To bounce exactly once off the left and right border means, on the expanded grid, to cross an odd and an even vertical line. That can only be done by moving 2 cells left or 2 cells rights.
The same applies vertically. You need to move 2 cells up or 2 cells down.
In order to get back to the exact same point, you need to move exactly 2 units horizontally and 2 units vertically. That means the x and y displacements are a multiple of the x and y table size (modulo the sign), which is equal to say the movement is parallel to a diagonal.[/hide]

Title: Re: Billiard Ball Problem
Post by Hippo on Nov 22^nd, 2007, 5:25am

Just 2 corrections:
Parallel to A diagonal of the table REDUCED BY HALF the ball DIAMETER from all sides. ...

Rest of the reasoning is OK ;)

Title: Re: Billiard Ball Problem
Post by Grimbal on Nov 22^nd, 2007, 5:46am

And of course, the border of the table is to be excluded too ;).

Title: Re: Billiard Ball Problem
Post by Hippo on Nov 22^nd, 2007, 6:15am

on 11/22/07 at 05:46:34, Grimbal wrote:

And of course, the border of the table is to be excluded too ;).

Absolutely ;D, I ignore everything except the place where the balls can be. But It seemed to me that specifying the diagonal we are talking about may be important as a lot of people sees another one ;) when not stated exactly.

Title: Re: Billiard Ball Problem
Post by towr on Nov 22^nd, 2007, 7:29am

My billiard balls have a radius of zero, the same as the width of my tables borders ;)

Title: Re: Billiard Ball Problem
Post by Hippo on Nov 22^nd, 2007, 12:40pm

on 11/22/07 at 07:29:57, towr wrote:

My billiard balls have a radius of zero, the same as the width of my tables borders ;)

Funny game :D isn't it ... just to hit such a ball with a cue :)