Amber G. wrote:<snip>
For the Yugoslav problem, the answer is 2, (That is rabbit escapes if it can run 2x (or more) times faster). As gakkakad mentioned, the answer is by no means trivial.
From the diagram above, r1 is the initial position of the rabbit, f1 is the initial position of the fox. The base of the triangle represents the X axis.
The fox adopts the following strategy -
If the rabbit moves away from r1 towards r2, the fox moves in a direction perpendicular to the direction of the rabbit i.e. from f1 towards f2.
Let us say at r2, if the rabbit reverses and moves towards r3, the fox moves in a direction perpendicular to the opposite side of the triangle i.e. from f2 towards f3.
In both cases, the fox moves with a minimum speed such that its x coords always matches the x coords of the rabbit.
The fox will always keep moving closer to the rabbit along the y axis until it finally catches it (the distance can increase when the rabbit is reversing, however for a given point of reference, it will keep reducing e.g. if the rabbit backs up all the way to its starting point, the fox would have been closer than it had been earlier). If the rabbit keeps still, the fox moves along the y-axis
What remains is to trivially determine this minimum speed at which the x coords remain equal. If the rabbit keeps going to the left towards e, the fox will catch up with it if it travel at 1/sqrt(3) times the speed of the rabbit. Motions in the opposite direction are similar i.e. the fox always has to travel at 1/sqrt(3) times the speed of the rabbit so that the x coords match while it comes closer along the y-axis until it finally catches up with it.
So, if the aim is for the fox to catch the rabbit, the rabbit does not require to travel @ less than 2x but only at less than sqrt(3)x
. To put it another way, the fox is guaranteed to catch the rabbit if it is faster than 1/sqrt(3) times the rabbit. Indeed, it is trivially possible to calculate the angle at which the fox should travel for a given ratio of speeds.
I think this problem as given is simpler than the duck-fox lake problem but it might be tougher to prove the minimum ratio at which the duck can never be caught, especially for the range between sqrt(3) and 2.