Source: Moscow Puzzles
Problem:
A soccer fan upset by the defeat of his favorite team slept restlessly. In his dream ,he saw the goalkeeper was practicing alone in a large room, tossing the ball against a wall and catching it.
But the goalkeeper got smaller and smaller and changed into a ping pong ball, and the soccer ball swelled into a huge cast iron ball.
The iron ball circled around madly, trying to crush the ping pong ball which darted desperately about .
Could the ping pong ball find safety without leaving the floor?
The answer given by all is :the ping pong ball should place itself in one of the corner of the room, so the large ball can’t touch it.
ReplyDeleteBut everybody assumed some radii for both the balls, so they can’t touch.
There must be a relation between the radii of the balls ,so that crushing is not possible.
For crushing, the balls must touch each other
And for not touching, they must maintain a distance of ‘k’ between them.(the distance shown by the arrows)
Lets radius of large ball is R and that of small ball is r.
For both the cases ,when the ball touches and does not touches the equations are:
Touching:::R^2 +(2r)(2r+2R) = (R+2r)^2
Not touching:: R^2+(2r+k)(2r+k+2R) =(R+2r+k)^2
So (R+2r+k)^2 > R^2 +(2r)(2r+2R)
K >(R^2 +4rR)/(4r+2R)
If this condition is satisfied, then also the ball can roll on the floor without being crushed.