How far can a dog run into the woods? 

All the way.

Any given woods has a center; any dog running from the outside to the center can be considered to be running "into" the woods, and any dog fleeing the center can be considered to be running "out of" the woods. So the maximum distance a dog can run into the woods is from the edge to the center, and, hypothetically, a dog could run that entire distance.

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re: How far can a dog run into the woods? 

“It occurs to us that the dog’s continued presence there would be useful, in case something happens to that tree that we’ve been hearing about since high school physics — the one that falls when no one is in the forest and since there is no eardum to register sound waves, makes no noise. You know what a fine sense of hearing a dog has. Let him run [all the way in], settle there, and keep an ear cocked for that tree.”

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