Author |
Topic: Billiard Ball Problem (Read 678 times) |
|
cool_joh
Guest
|
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?
|
|
 IP Logged |
|
|
|
cool_joh
Guest
|
 |
Re: Billiard Ball Problem
« Reply #2 on: Nov 22nd, 2007, 1:03am » |
 Quote  Modify
 Remove
|
on Nov 21st, 2007, 11:25pm, towr wrote:| It seems, parallel to the diagonal |
|
How can you prove it?
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender: 
Posts: 13730
|
|
Re: Billiard Ball Problem
« Reply #3 on: Nov 22nd, 2007, 4:36am » |
Quote Modify
|
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.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: Billiard Ball Problem
« Reply #4 on: Nov 22nd, 2007, 5:08am » |
Quote Modify
|
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.
|
|
IP Logged |
|
|
|
Hippo
Uberpuzzler
Gender:
Posts: 919
|
|
Re: Billiard Ball Problem
« Reply #5 on: Nov 22nd, 2007, 5:25am » |
Quote Modify
|
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 
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: Billiard Ball Problem
« Reply #6 on: Nov 22nd, 2007, 5:46am » |
Quote Modify
|
And of course, the border of the table is to be excluded too .
|
|
IP Logged |
|
|
|
Hippo
Uberpuzzler
Gender:
Posts: 919
|
|
Re: Billiard Ball Problem
« Reply #7 on: Nov 22nd, 2007, 6:15am » |
Quote Modify
|
on Nov 22nd, 2007, 5:46am, Grimbal wrote:| And of course, the border of the table is to be excluded too . |
|
Absolutely  , 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.
|
| « Last Edit: Nov 22nd, 2007, 6:15am by Hippo » |
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Billiard Ball Problem
« Reply #8 on: Nov 22nd, 2007, 7:29am » |
Quote Modify
|
My billiard balls have a radius of zero, the same as the width of my tables borders
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Hippo
Uberpuzzler
Gender:
Posts: 919
|
|
Re: Billiard Ball Problem
« Reply #9 on: Nov 22nd, 2007, 12:40pm » |
Quote Modify
|
on Nov 22nd, 2007, 7:29am, towr wrote:| My billiard balls have a radius of zero, the same as the width of my tables borders |
|
Funny game  isn't it ... just to hit such a ball with a cue 
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print