<t>Well,If we assume the fall of a needles uniformly random, let assume that a needle position is represented by an absciss x and an angle theta ( we obviously don't care about y). x and theta have uniform distribution. You integrate 1 on the plane (x,theta) where the the needle crosses a line:theta...

instead of , let s use theta as the angle with the y axis distance = 2 * cos (theta)gravity = g * cos (theta)time = 2*sqrt ( distance/gravity) = 2/sqrt(g)

As a very quick answer you can imagine that in average, 10 ordered [0,N] uniform variables would be spaced by N/11so the max one would be 10/11*N. Thus 10/11*N=450 => N= 495