An Excellent Puzzle

A friend of mine of scientific mind wanted to invent a puzzle where possession of a fine chronograph, but nothing else, would be the key to its solution, so he rang me up and we settled on a kidnapping/desert island scenario as having the greatest promise for a chronograph to be the hero of the puzzle, only, as we honed the puzzle to plausibility and perfection, the chronograph became utterly superfluous. So we took it out of the puzzle, and left the victim truly stranded, but with a superbly elegant escape. Here it is. See if you can find the solution:

It is a dark and stormy night in Detroit in the dead of Winter. You're the son (or daughter) of a wealthy celebrity with intelligent friends, and everything goes suddenly blank when you're blackjacked on the back of the head and kidnapped. When you come to, it is still night, but the weather is tropical, and the stars crowd a cloudless sky from horizon to horizon. You struggle to sit upright on bare sand, less than a hundred feet away from a shore washed by a gentle surf. In front of you, a dark figure looms, and drops something blocky and heavy at your feet.

"That's a satellite phone," the figure says, his voice hard but calm. "Wait 'till daylight, then call your family; prove you're alive. You went missing two days ago. Be brief; there're only about 60 seconds of battery - not enough time to get a fix on you." The figure pauses to light a cigarette, then continues. "We left you some water and sanwiches," he says, gesturing to a small paper sack a few feet away. "If we collect our ransom, we'll come back and take you to an airport. If we don't, there are no second helpings. Any questions?"

"How much?" You ask weakly.

"Eight million."

"More than my allowance. Where are we?"

"A long way from Detroit. There isn't so much as a boat dock within a hundred miles."

"I can't swim anyway."

"We know."

Without another word, the figure turns away and pushes off into the surf in a small motor skiff. A mile offshore, barely discernible in the darkness, the blacked out hull of a larger vessel eclipses the stars near the horizon, and the wake of the fast disappearing motor skiff casts a faint phosphorescent arrow straight toward it. Less than an hour later the skiff and its mother ship have vanished.

You quickly assess your situation. The island you're on is completely featureless, flat, and less than a mile in both width and breadth. Bits of seaweed are its sole vegetation. You too are similarly stripped; the kidnappers left you clothed in light shorts and shirt, pockets empty, your watch and other jewelry gone. They did leave you a baseball cap for protection against the sun when daylight arrives. The bag of water and sandwiches is enough to last only a few days, if rationed.

Your family has the resources to rescue you if you could communicate your location, but there's no way for you to determine that. You never took any interest in astronomy, for example, so the pattern of the stars is as meaningless to you as the pattern of waves against the beach.

But before sunrise you realize you CAN communicate your location, and with precision, without ever knowing yourself where you are. And the 60 seconds of satellite telephone time is ample. What do you do?

 

Congrats to Brick, he (she?) nailed it beautifully, and quickly too!

No one we've ever presented this puzzle to has ever gotten it without lots of mental nudging, so Brick's rapid, complete answer deserves a prize. Next time you're out SFO way an airplane ride and a beer are on me.

The only two days of the year the victim would be screwed are the equinoxes, so that's why it's set in deep Winter. The sunrise sunset calls do indeed fix longitude, but more important, they describe great circle arcs that intersect. The time interval between calls fixes latitude, a third great circle arc that forms a small triangle of intersection with the sunrise sunset arcs. The victim is inside that triangle.

Other details like the wake and the baseball cap are the standard red herrings stashed in any puzzle, to keep you guessing as to what's really important and what's chaff.

Good job Brick!

MB

 

<div class='quotetop'>QUOTE(FL_Prius_Driver @ Oct 1 2007, 08:10 PM) [snapback]520124[/snapback]</div>

One of our original answers was considerably convoluted and involved ascertaining which side of the equator the island was on (and other complicated gyrations I won't get into but are surely why Tom and Ray rejected the puzzle when first sent to them). Identifying north or south hemisphere is simple: your shadow points to the hemisphere you're in. Stand facing a rising sun and you're facing east. Stand long enough for your shadow to angle away from an east-west line and where it points is the hemisphere you occupy.

Fixing north or south is simply a matter of watching your shadow shrink (our original answer involved poking holes in the sand at the extremities of a water bottle's shadow throughout the course of the day) - where it's at its smallest is its north-south orientation.

I wish I could claim credit for the elegant answer Brick so succinctly identified, but alas, my mind is not so nimble. We were presenting the puzzle and our original answer proudly to a colleague, who listened patiently at first, but finally, exasperated, asked why the hell not just make two phone calls and save a whole lot of useless futzing around? He was absolutely correct, of course, which is why the correct answer is so elegant.

Mark Baird
Alameda CA