A lone chess bishop finds himself standing on the (zero-based indexing) square (x, y) of a curious m * n chessboard floating in outer space, covered with completely frictionless magic ice. This board is not necessarily square, but can be any rectangle of integer dimensions. The board is surrounded from all sides with bouncy walls similar to an air hockey table. Some of the squares on the board contain aliens that make game over man for anyone who enters that square.

The bishop can initially propel himself to any of the four diagonal directions. After this initial impulse gets him going, the bishop can no longer control his movements on the frictionless ice, but will keep sliding towards whatever end the present direction vector and bouncing from the walls take him. It is even possible for the bishop to forever repeat the same cycle of squares.

The bishop must reach any one of the four possible exits at the four corners of the board. This function should determine whether there exists at least one starting direction that leads the bishop to the safety of any one of the four corners, avoiding all the aliens along the way. Look at the example x = 3, y = 2, m = 4, n = 5, aliens = [(2, 2), (1, 4)] => True.

Input: Four integers (int) and list of tuples of two integers.

Output: Logic value (bool).

Examples:

assert reach_corner(0, 2, 5, 5, []) == False
assert reach_corner(4, 4, 9, 9, [(0, 0), (0, 8), (8, 0), (8, 8)]) == False
assert reach_corner(1, 1, 1000, 2, [(0, 0), (0, 1), (999, 0)]) == True

Preconditions:

• 0 <= x < m, 0 <= y < n;
• m > 1, n > 1;
• (x,y) not in aliens.

The mission was taken from Python CCPS 109. It is taught for Ryerson Chang School of Continuing Education by Ilkka Kokkarinen