REALLY HARD GEOMETRY/ALGEBRA QUESTION!!! HELP!!!?

Question

Perplexing Paula Pocket is a pool shark. Besides being a great pool player, she makes pool tables. The pool tables are indeed perplexing. They are all different size rectangles with only four pockets. The pockets are at the four corners of the tables.

A __________ B

| |

| |

| |

| |

| |

|_________|

C D

What makes these tables most interesting is the different games she plays on them. She picks a table and lays the cue ball right in front of pocke C. She challenges her opponent to pick the pocket where the cue ball will drop. Then she always strikes the ball at a 45 degree direction to the sides of the table, The cue banks of different sides of the tablke until it finally drops in onepocket. If you just guess, you have a 1 in 4 change of guessing right. But then she also asks you to predict the number of times the ball will bank off a side before dropping in a pocket.

Remember she has almost an unlimited number of tables with different dimensions. Except Paula doesn't like fractions so all her tables have whole number dimensions in feet.

You want to be able to beat Paula at her own game. Determine a strategy that you can use to play with Paula. The strategy would be that when Paula picks a table and give you the dimensions, you would be able to accurately predict the number of times a cue ball would bank off the wall and exactly which pocket the ball would drop.

Explain how you go your answer

THANKS :D

Answer 1

Okay, Paula picks some table and gives you the length(l) and the width (w).  Now, suppose that you're looking at the table this way:

_ _ _

|_|_|_|

|_|_|_| length

|_|_|_|

width

Let's call 'a' the number of times the ball goes across the width of the table and 'b' the number of times the ball goes across the length of the table.

Since the pockets are at the corners, we know that 'a' and 'b' must be positive integers.

Also, since the ball will bank off a side every time it crosses either the width or the length of the table EXCEPT the last time, we know that a + b - 2 will be the number of times the cue ball will bank off a wall.

How do we find a and b?

Well, we know that if the width is twice the length, the ball will cross the length twice and the width only once.

In fact the ratio will always be the same!

That is, w / l will always be the same as b / a.

w / l = b / a

aw = bl

Now, since the ball is always going to go into the first pocket it reaches, aw and bl must be the least common multiple of w and l.

aw = bl = lcm(w, l)

So, we can find the number of times the ball banks off a wall:

lcm(w, l) / w + lcm(w, l) / l - 2

Now, we just need to find which pocket the ball lands in.

If the ball crosses the width once (a = 1) then it will be at the right side.  If the ball crosses the height once (b = 1) then it will be at the top side.  Every time you add one more, it switches sides.

So, in general,

a is even => left side

a is odd => right side

b is even => bottom side

b is odd => top side

Using this information, you can now determine which  pocket (bottom left, top left, top right, or bottom right) the ball will land in.

Answer 2

In response to the above solution,

what I don't understand is why does lb=wa equal the least common multiple?