## 4 dogs chasing on corners of square

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Puzzle:
There are four dogs, each at the counter 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 continously adjust their direction so that
they are always heading straight towards their clockwise neighbor. How long does it take for the dogs to
catch each other? Where does this happen? (Hint: Dog’s are moving in a symmetrical fashion, not along the edges of the square).

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.

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