Chasing Dogs.

There are four dogs, each at a corner of a large square.
Each of the dogs begins chasing the dog clockwise from it.
All of the dogs run at the same speed. All continuously
adjust their direction so that they are always heading
straight toward their clockwise neighbor. How long does it
take for the dogs to catch each other? Where does this
happen?

Solution :

To make things easy, let’s say the square is 1 mile on each
side, and the dogs are genetically enhanced greyhounds that
run exactly 1 mile per minute. Pretend you’re a flea riding on
the back of Dog 1. You’ve got a tiny radar gun that tells you how
fast things are moving, relative to your own frame of reference
(which is to say, Dog l’s frame of reference, since you’re holding
tight to Dog l’s back with five of your legs and pointing the
radar gun with the sixth). Dog 1 is chasing Dog 2, who is
chasing Dog 3, who is chasing Dog 4, who in turn is chasing
Dog 1. At the start of the chase, you aim the radar gun at Dog
4 (who’s chasing you). It informs you that Dog 4 is
approaching at a speed of 1 mile per minute.
A little while later, you try the radar gun again. What
does the gun read now? By this point, all the dogs have
moved a little, all are a bit closer to each other, and all have
shifted direction just slightly in order to be tracking their
respective target dogs. The four dogs still form a perfect
square. Each dog is still chasing its target dog at 1 mile per
minute, and each target dog is still moving at right angles to
the chaser. Because the target dog’s motion is still at right
angles, each chasing dog gains on its target dog at the full
running speed. That means your radar gun must say that Dog
4 is still gaining on you at 1 mile per minute.
Your radar gun will report that Dog 4 is approaching at
that speed throughout the chase. This talk of fleas and radar
guns is just a colorful way of illustrating what the puzzle
specifies, that the dogs perpetually gain on their targets at
constant speed.
It makes no difference that your frame of reference
(read: dog) is itself moving relative to the other dogs or the
ground. One frame of reference is as good as any other. (If
they give you a hard time about that, tell ‘em Einstein said
so.) The only thing that matters is that Dog 4 approaches you
at constant speed. Since Dog 4 is a mile away from you at the
outset and approaches at an unvarying 1 mile per minute,
Dog 4 will necessarily smack into you at the end of a minute.
Fleas riding on the other dogs’ backs will come to similar
conclusions. All the dogs will plow into each other one
minute after the start.
Where does this happen? The dogs’ motions are
entirely symmetrical. It would be strange if the dogs ended
up two counties to the west. Nothing is "pulling" them to
the west. Whatever happens must preserve the symmetry of
the original situation. Given that the dogs meet, the collision
has to be right in the middle of the square.