In this problem we are dealing with rectangular pool boards. Our boards have pointwise holes at the corners and nowhere else. A pointwise pool ball is running on the board and it never stops but falls down at a corner when it arrives there. If the ball starts its run at the left bottom corner at angle 45o to both of the adjacent sides of a 4×6 board then its path is shown below.
(I.) To which corner the ball arrives?
(II.) What is the total number of cushions until it falls down at the corner?
(III.) What is the total length of its path? (Through how many 1×1 squares does it go through?)Try to answer these questions in the following cases:
a) 3 × 5;
b) 4 × 8;
c) 15 × 18;
d) 2021 × 235;
e) 2021 × 900.f) Try to answer questions (I.) , (II.) , (III.) in the case of an m×n board in terms of m and n where m and n are positive integers.
I have got no idea how to solve any of these!
Would you mind telling me how I can begin to understand and solve this question?
This is not an answer, more like a remark that may or may not be helpful. It's a way of diagramming the problem.
Consider the case of a billiard table $6$ units long and $4$ units high. In the following figure every point where the $45^{\circ}$ line crosses one of the dotted lines corresponds to a bounce off a cushion, so the ball bounces off $3$ cushions before dropping into a corner pocket.