**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:**

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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