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Pool table investigation.
By Kane Limbrick (P1960) on Saturday, July 8, 2000 - 08:13 am:

Hi,
I need a formula to work out,on a pool table of any size, how many moves a ball travelling at a constant speed will take to find a pocket in any 4 of the table's corners. A move is until the ball hits the edge of a table and then a new move begins as it bounces off. The ball starts in the bottom left-hand corner travelling at 45 degrees.

Thanks, Kane.

By Neil Morrison (P1462) on Saturday, July 8, 2000 - 11:26 am:

Firstly, the ball will also make an angle of 45 after it hits the cushion, because of the angle of incidence idea (unless there is spin on it)

By Kane Limbrick (P1960) on Saturday, July 8, 2000 - 01:06 pm:

Yes I know, but I need a formula.
e.g Length = 2m Width = 1m Moves = 2
Length = 3m width = 1m Moves = 3

Length = 4m Width = 3m Moves = 6
Length = 5m Width = 3m Moves = 7

By Neil Morrison (P1462) on Saturday, July 8, 2000 - 02:32 pm:

Well in that case a simple pythagoras will suffice.

Neil M

By Dan Goodman (Dfmg2) on Saturday, July 8, 2000 - 04:44 pm:

Here's a way to think about this problem, draw a large grid with the dimensions of your table, now draw a line at 45 degrees from one of the corners of your grid. When this line passes from the first rectangle into the second, imagine that you have flipped the rectangle over on that edge, now if you continue the straight line, this is (in a sense) the same path the ball would be going along if it had bounced! Maybe a diagram will clear up what I'm trying to say:

Pool table investigation

In this diagram, lines with the same colour correspond to one another. From this diagram, you should be able to see that if the ball is in a corner of the grid on the long straight line, then it must also be on the corner of the ball which stays within the grey rectangle. This should help to find a formula.

By Simon Judes (P2636) on Saturday, July 8, 2000 - 05:14 pm:

It occured to me, the formula might be quite strange; because the angle the ball goes off at, and the lengths of the sides, are continuous variables, whereas the number of crossings on the grid is countable. So there might even be values of the parameters for which the line never hits a crossing in the grid.
Perhaps you can solve this by specifying some tolerance, i.e. the size of the pockets.

By Dan Goodman (Dfmg2) on Sunday, July 9, 2000 - 12:53 am:

There'll always be a finite answer if we consider only lengths whose ratio is rational, not so if we allow irrational ratios, e.g. sqrt(2).