How do you write an algorithm that animates a circle bouncing around a
window.  Using the change in x and y to control the movement.  Using a
loop and each time you go through the loop the x and y and move + or -
1.  If the circle hits the edge then the sign changes to - from + or
vice-versa.

Using the pygame library, this is fairly easy, here is an example.

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import sys, pygame
pygame.init()

size = width, height = 600, 400
speed = [20, 20]
black = 0, 0, 0

screen = pygame.display.set_mode(size)

ball = pygame.image.load("ball.gif")
ballrect = ball.get_rect()

while 1:
    for event in pygame.event.get():
        if event.type == pygame.QUIT: sys.exit()

    ballrect = ballrect.move(speed)
    if ballrect.left < 0 or ballrect.right > width:
        speed[0] = -speed[0]
    if ballrect.top < 0 or ballrect.bottom > height:
        speed[1] = -speed[1]

    screen.fill(black)
    screen.blit(ball, ballrect)
    pygame.display.flip()
------------------------

Make sure you have an image (named ball.gif in this example) in the
same directory as the script.

Hope this helps,
Tom