Maximize the zigzags

Imagine that in order to reach your destination you would normally have to walk along busy streets a mile east and then a mile north. Now, if you were driving, you'd simply take the two streets and you'd be there. Terrific. But since you're walking, you need a faster way.

Chances are there is a grid of lesser-used side streets between your starting point and your destination. Perhaps these streets have quaint homes on them instead of strip malls. Perhaps there are trees with sweet-sounding singing birds in them instead of asphalt and engines. Perhaps there are charming parks or school playgrounds to cut through as well.

(On the other hand, perhaps these side streets lead to neighborhoods which are a little rougher than you'd feel comfortable walking through -- you may want to scope your route out first.)

If you start walking east and turn left onto the first available side street that goes north, and then turn right at the end of the block onto the first street that goes east, you are on your way to maximizing the zigzags.

Now, at first glance, it's evident that you are going to be walking the same distance east and the same distance north as you would have if you had taken the main streets, you're just going to be alternating between directions, traveling a little each way at a time.

This would be true except there's much less traffic on side streets. Instead of walking on the sidewalk on one side of the street like you would have had to do if you had taken the main streets, you can cut diagonally across the side street. This is where the benefit of the hypotenuse comes in. If you start at the southwest corner of a quiet side street and walk unimpeded to its northeast corner, you've saved the difference between the square root of the sum of the squares of the side street's length and width and the sum of the side street's length and width. This done repeatedly until you arrive at your destination can amount to a huge savings in time!