the crazy billiard table
A man designs a 2m by 2m billiard table with only 4 pockets, one in each corner, marked A, B, C, D. A ball is shot from directly in front of one pocket at a 45 degree angle. Obviously, because the billiard table is square, it will go into the diagonally opposite hole.
What happens now if the table is increased to 2 * 3, 2 * 4 , 2 * 5 etc. and the ball is again hit at a 45 degree angle. Is there a way to predict what pocket the ball will go into in each trial?
What about for a table 3 * 3, 3 * 4 3 * 5 etc. Can you predict what pocket it will go in?
Is there a way to write a generalised formula for this, to find out which pocket it will go into for a very large table, 342 * 275 for example?
use pythagoross theorem I attach some pictures hope that will help you
Originally Posted by bruxism
begin from the two green arrow blue numbers for the segments on the sides which the ball did
for the 342 * 275 table I draw I complex pic here I use orange numbers to determine the new segments after the ball hit the four sides
Is there any questions ?
Actually quite "easy" to draw the paths; above is a 10 by 22 table ABCD; cue ball shot from A:
D 2 G 8 C
1: hits BC at E (10 above B)
2: hits AD at F (2 below D)
3: hits CD at G (2 from D)
4: hits BC at H (8 from C)
And you're basically finished: you have 4 lines (AE, EF, FG, GH):
ALL other paths will form lines parallel to these lines; continuing:
from H simply draw a line to side AD parallel to line FG.
And carry on similarly until a path line hits one of corners.
In this example, corner C will be hit after 14 bounces off the sides.
If you really want to have fun, get a sheet of graph paper and draw
a table 27 by 34 (easy enough using the little squares as units).
Go as I explained above: this time you'll hit corner D, after 59 bounces!
Haven't tried for a general case formula (probably won't either!), but
obviously has something to do with LCM(long side, short side).