Philip

Neglecting inertia and response time of the lion, The tamer would always be caught if he is required to change direction (regardless of the shape of the cage). because each time the tamer changes with even the slightest direction at the slightest angle, the lion will be able to switch to the shortest direction to where the tamer is (even if blindly following him).

If we consider only integers, from zero to infinity, and we could list them all, we could see that the quantity of low-value-numbers is negligible when compared to the quantity of very-large-valued numbers. Thus we can safely assume that when randomizing, we will pick only very-large-valued numbers. It is also said that randomizing 0-9 and combining them to form enormously (infinitely) large number, every combination of strings (including ..7777777..) is contained in that number. So, since we can safely assume that we will be able to pick only very-large-valued numbers, and in each of those numbers, the string 7777777 is contained, We can assume that every time we pick a number randomly, the string 7777777 is contained in that number. Thus, 100%. Also the same for negative numbers. And it won't matter if they are integers or not, you can use the same idea for all rationals.

If without air resistance and not on North or south pole, parabolic, because of the initial velocity given by the spin of the earth while still on hand, similar to throwing the ball with some initial velocity, and actually seeing the parabolic projectile shape. if on North or south pole, linear. If taking into account the speed of the motion of earth around the solar system (revolution), the rotation of earth would be negligible, thus it is near elliptical (same with galactic reference of sun orbiting galaxy).

Connecting the 8 corks with web with least linear meter requires connecting with the web along the sides of the cube (no diagonals) with length 1m, and uses at least 7 segments to connect 8 corks. so the total length is 7*1m = 7 meters. and the shape could vary, as long as you pass the web only along the sides of the cube.