Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."
In her Parade Magazine column of February 15, 1998, Marilyn claims that you could see a 150 foot tall lighthouse from anywhere on a five mile long, five mile wide desert island.
If the island is approximately square, the lighthouse could be as much as 7 miles away, if situated on a corner, or 3 1/2 miles if at the center. (If we knew where it was, we'd know where it was.) Let's consider the case where it could be, say, 5 miles away. There are various approaches, but a general feel for its visibility can be obtained by dividing the height of the lighthouse (150 feet) by its distance (5 miles or 5 x 5280 feet, or 26400 feet). It comes out to about 1/176 radian or 1/3 degree--less than the diameter of the full moon, above the horizon. Even this assumes that the curvature of the surface of the earth has no effect. At 5 miles, the curvature is enough to lower this by .07 degrees, making it closer to 1/4 degree, or one half the full moon's diameter above the horizon.
Is this low enough to be blocked by sand dunes? Certainly! Again, even without the curvature of the earth's surface, a 15-foot dune 1/2 mile away mile will hide a 150-foot lighthouse 5 miles away.
In this puzzle, we thought we were working on a classic liar/truth-teller logic puzzle; once again, Marilyn throws in some lateral thinking (which is commendable when done right), but then gives an implausible answer, robbing us of the logic problem answer we thought we'd get.
The answer Marilyn gave is outrageous for such a simple problem, to which many assumptions can and must be added.
Marilyn assumes all desert islands are perfectly flat, so that a 150-foot lighthouse is visible for five miles on any of them. San Diego has an ancient lighthouse also (active 1855 to 1891), less than a mile from the active one on Point Loma (active 1891 to present). Neither lighthouse is visible from the top of the other!.
The old lighthouse was built on top of a 422 foot hill, too high to be consistently seen because of fog (the plane of the light is 462-ft above sealevel). The new one was built at the base of the same hill which is quite flat on top. The edge of the hill obscures the view in both directions. If you're ever in San Diego, the old lighthouse at Cabrillo National Monument is a good place to visit. I've heard that it was at one time, the most visited National Monument in the country.