In this problem, the six bugs start at the corners of a regular hexagon (with side length=10 inches).

Again, the bugs travel directly towards their neighbor (counter-clockwise). And, again, each bug homes in on its target, regardless of its target's motion. So, their paths will be curves spiraling toward the center of the hexagon, where they will meet.

What distance will the bugs have covered by then, and how did you determine it?

Similar to how Charlie solved the 3-bug problem...

Let's consider the vector diagram of a pursuer and pursuee pair of bugs.  Calling each vector V, the component of the pursuee's motion that is parallel to the pursuer's is 1/2*V AWAY from the pursuer.  So this is a 2:1 ratio as well, but now the pursuee is moving away from the pursuer instead of towards it.

In the 3-bug problem, you can solve for the distance travelled by saying "distance of the pursuer" + "parallel distance of the pursuee" = 10, and using the ratio found you get x + 0.5x = 10

In this case, we still have x and 0.5 x, but it is now x - 0.5x = 10.  If just changing that sign doesn't sit well with you, think of it this way: The pursuer needs to travel the distance of the edge, PLUS the distance the other bug runs away from it.  So x = 10 + 0.5x, which is the same thing.

So x = 20 inches.