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The new land surveyor and his boss arrived at the work site, a huge and perfectly flat parking lot. His boss said, "Your first assignment is going to be a tough one. You see this parking lot? It has light posts every 50 feet east to west and north to south."

"A perfect Cartesian grid with a light post at each intersection?" asked the Laziest Surveyor.

"That's right. And too many posts to count. They seem to go on forever."

"What's the job?" asked the Laziest Surveyor.

"They're building... something. I don't know what, but it's going to be crazy looking. Just look at the polygon on this map. Angles going every which way! That's the footprint of the new building. Must be one of these Deconstructivist architects or something. Not my style. Give me a nice rectangular Parthenon any day. But never mind that. Our job is to measure the area of that polygon."

The Laziest Surveyor glanced at the map. "Well, at least none of the sides of the polygon cross over each other. And none of the posts lie on the border except at the vertices. That should make things easier."

"You're an optimist!" the boss laughed. "This thing has 21 vertices, each one of them located exactly at a light post. There's convex angles, concave angles, peninsulas, coves, what a mess. I hope you know your trigonometry! This will take you all day."

"When construction begins, I guess they'll have to knock down these lights," said the Laziest Surveyor, looking all around him.

"Just the ones inside the polygon. They'll leave the ones at the vertices up. Let's see, so they'll bulldoze..." The boss flipped through work orders as the Laziest Surveyor stared expectantly. "Wow, 149! Big job! Well, you've got a lot of work to do so I'll leave you to it. Here's the map and the survey form."

The boss drove away. As soon as the boss' car was out of sight the Laziest Surveyor crumpled up the map and dropped it on the ground. He took out a pocket calculator, jabbed at it a few times, and wrote the correct area on the survey form. Then he spent the rest of the day at a baseball game.

What area did he calculate and how did he do it?

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He calculated the area using Pick's Theorem.

Area = (number of internal grid points) + (half the number of boundary points) - 1

The area is $149 + \frac{21}{2} - 1 = 158.5\mbox{ units}^2$

Since we're given that the units are $50\mbox{ ft}$, the area is $396250\mbox{ ft}^2$.

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    $\begingroup$ Even without Pick's theorem, I would draw some arbitrary shape with 21 vertices and 149 light posts but an easily-calculable area (e.g. add repeating "saw-teeth" for many of the vertices), and figure that if the area is correct for that and the puzzle is solvable, the building in the puzzle must have the same area. $\endgroup$ – supercat Nov 4 '14 at 21:41

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