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Math Help - the crazy billiard table

  1. #1
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    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?
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by bruxism View Post
    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
    begin from the two green arrow blue numbers for the segments on the sides which the ball did

    the crazy billiard table-good.jpg


    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

    the crazy billiard table-goood.jpg

    Is there any questions ?
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  3. #3
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    Code:
    D 2 G    8       C
    2
    F                8
     
                     H
                     4
     
                     E
     
     
                    10
     
     
    A                B
    Actually quite "easy" to draw the paths; above is a 10 by 22 table ABCD; cue ball shot from A:
    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).
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