A solution via facebook was proposed and I want to share it here:
Димитър Марков It is just a GUESS based on the fact that each insect will move towards its chosen one. And since the chosen one will not stand still the chaser will move not only ahead but a little bit to the "right" in each moment. It seems to me reasonable the trajectory to be a circle with center the middle of the side AB (for instance) and radius 5. To the center each insect will travel a quarter of the length of the circumference. I'm trying to proove this. Got apples?
I replied with few hints I want to share here, so the race to be fair!
Your calculations are absolutely right, but having such a scheme as you described means that the bug A path will be generated by two directions only – up and right (I mentioned A, but the case for the other three bugs is the same, just using two other directions).
Are you sure that will be the case? Are you sure the curve will behave like that? I can give you a hint – the curve will not behave like that. The curve will use all the four directions. No matter what curves we are going to draw, using only 2 or 3 directions to generate their path – will just approximate the solution to a specific point when the rest of the directions should participate. If you start from point A of the square moving towards the center of the square – using for example only Up and Right directions would be a wrong approach. Furthermore the private case you have stated (a quarter from a circle) is just only one such possible drawing from the infinite set of possible cases.
My last hint will be that the answer is greater than 2.5 * pi.
The answer is an integer number and as such pi is not involved.
Excellent thought so far, just you need to dig a little bit deeper!
Thanks for your answer!
Let’s apple ourselves!