Puzzles for Programmers and Pros

6.4. Picturing the Treasure

The previous three puzzles all had a generic, though inefficient, solution strategy: Try all possible rearrangements and see if any meet the constraints. This one demands geometrical insight.

A pirate's trunk filled with millions of dollars worth of jewels is buried in the sand on a flat part of the deep ocean floor. Your client has dropped sensors in the area to try to locate the trunk. The sensors can be dropped to a precise location, but their reports of distance to the treasure trunk are accurate only to within 10 percent. The first sensor reports the treasure to be at a distance of 450 meters (so the real distance from that sensor falls between 405 meters and 495 meters). A second sensor reports the treasure to be at 350 meters (real distance is between 315 meters and 385 meters). Because your client doesn't want you to hire your own ship, he tells you only relative positions of the sensors. Specifically, he tells you the first sensor is at position (0,0) and the second is at position (300, 400).

6.4.1. Warm-Up

If the sensor estimates of distance to the treasure were precisely correct (no plus or minus 10 percent), how could you use one more sensor to find the exact position of the treasure?

6.4.2. Solution to Warm-Up

Draw a circle of radius 450 around (0,0) and another circle of radius 350 around (300, 400). The two circles meet at two points, ...