I guess “pay the ransom and get my location” is a valid solution.
Brian
One. I misread the exact scenario
Two. This line from the OP messed me up
As pointed out easy IF you know you’re local time.
OK, not longitude, but a more or less N/S line
And I misread the map, if it is dawn, you are probably south of Mexico (or in Hudson Bay)
Brian
I wish I could offer more help for a solution I think the solutions here are very cool to ponder. I did however want to add:
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Very cool thought provoking riddle - thanks N9IWP
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This scenario strikes me as remarkably similar to an episode of the old British TV show “The Prisoner”. I recall Number 6 trying to use some similar method (of course without a cell phone) to determine the latitude and longitude of the The Village. Ring any bells?
Maybe the Riddler is a Prisoner fan?
Yeah…
Better hope you didn’t sign a “No-ransom” agreement.
I thought every Boy Scout, experienced puzzler, or astronomy buff already knew this trick, but I suppose not, so I’ll post it:
You don’t need to estimate the length of a shadow to find noon. You plant an object with an obvious tip in the sand, standing vertically. A water bottle will do. At some point in the day, at any time, you mark the shadow’s tip with any object. Another water bottle will work, especially if laid on its side, but I’d prefer a seashell or pebble.
Then you wait at least 15 minutes, but preferably several hours. You then mark the tip again. Connect the two marks with a line. Wait a few hours more and verify with a third point.
This line is a direct east/west line. It’s not an arc. The shadow changes length as the sun moves through the sky, such that the shadow traces a perfect line. Your first point is West. Your last point is East. That gives you North and South as well. Whatever point on the line forms a right angle with a line to the original “gnomon” water bottle is noon. If the line is to the North of the bottle, you’re in the Northern hemisphere. If South, then Southern.
No need to measure shadows. No need to know the length unless you want a better estimate of longitude. But this technique is simple, quick, and gives you both time, direction, and hemisphere. You can get all this information, plus map the entire island, in under an hour.
Doesn’t help a lot for determining your location (other than hemisphere) and by the time you have determined noon it is probably past it…
Brian
I was just hit with some inspiration. How about manufacturing an astrolabe out of the water bottle?
- Drink the bottle of water so you stay hydrated.
- Refill it halfway with salt water. You coulda used the fresh water, but why?
- Tilt the bottle on its side, such that the water level is touching both the plastic notch in the center of the bottom and the center of the cap. This means you’re holding the bottle perfectly level, and therefore can “shoot” the horizon.
- Mark the water level on both the near side and far side with scratches or other permanent marker substitute. This gives a diameter.
Take a nap. You’ve earned it. Wait for noon.
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At noon, rotate the bottle (i.e., a barrel roll) such that the scratches point to the sun.
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While holding the scratches steadily in line w/ the sun, mark the new water level. You can mark both side,s but only one is necessary, and probably easier to read later.
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You now have two marks on a circle. The angle between the two yields the sun’s maximum angle of elevation. 90 degrees minus this angle is your approximate latitude, if you can adjust for the season.
Coupled with your stick-compass, rough sketch of the island, longitude from timing sunrise, and latitude from the astrolabe, you’re all set. Enjoy your beach vacation and await your friends’ arrival.
Not at all. You can determine noon any time 30 minutes after sunrise. You mark the shadow in the morning, then again 15 or so minutes later, and technically, you have a line.
You can project that line as far as you want. Whatever point on that line is at a right angle to the water bottle is noon, even if the current time is still 7:15 am.
Hell, you can project (heh) when sunset is by just mirroring your sunrise point (which is in the West) to the East side of your line. You have a whole clock in 15 or 30 minutes, and every extra shadow marking increases your accuracy.
After 4 markings (which takes one hour), you’re as accurate as you’ll need to be.
Slight correction to the astrolabe idea. Instead of trying to calculate my latitude from my measurement, I can just stop once I’ve measured the sun’s angle of elevation. My billionaire friends back home can punch that into a NOAA.gov calculator or whatever to do the seasonal adjustment. Hell, I don’t even need to know the current date because they already do.
For what it’s worth, and as my username might suggest, I’d give them a rough map of my island’s shape by naming squares on a chess board. If I say the square, it’s sand. If I don’t, it’s water. There’s even enough time to describe my system to them. An 8x8 grid ought to give them a good idea of what mile-by-half-mile island I’m on.
“If my island were on a chess board, it’d be shaped as…a8, b8, c8, d8…break…a7 b7…break…a6 b6 c6 d6…” and so on.
Guy Fleegman: I know! You construct a weapon. Look around you. Can you form some sort of rudimentary lathe? 
It would probably be of value to do a seconds count to see how long it takes the Sun to rise completely, from first sighting to fully above the horizon. Say this is 200 seconds. So now you know that when the sun is setting, from the time it grazes the horizon until it vanishes completely will also be about 200 seconds. This lets you time your phone call so that it will overlap that vanishing point; once you think the Sun is touching the horizon, wait 170 seconds and place your call. It’s easier to say when the Sun has disappeared than to say with certainty when it is 100% risen. The Sun takes from 2 to 5 minutes to rise and set depending on latitude and time of year, and it would be a waste to place your phone call too soon or late to capture the moment of disappearance.
Also, the folded paper bag will give you a 90 degree corner to help determine the point on your E-W shadow line that is closest to your shadow tower. That’s the noon point, and the ratio of tower height over shadow length at that point will help the rescuers find your latitude.
After you drink the first bottle of water, fill with sea water and turn it upside down.
If the draining water rotates clockwise, you are in the northern hemisphere. If it rotates counterclockwise, you are in the Southern Hemisphere.
If it can’t drain because the water can’t decide which way to rotate, you are either exactly in the equator to within feet, or you are starring in the Truman Show.
We are all assuming the goal is to be rescued without paying the ransom. But per the riddle, we are “super rich”. Maybe the solution is to wait for dawn, call our people and tell them to pay the fucking ransom.
I hope that was a joke, as the Coriolis force is VERY weak on something on that scale
Brian
Actually, that folded paper bag might work as a astrolabe too. Hold one corner toward your eye, with the adjacent 90 degree corner away from you. Holding your head still, manipulate the bag until you are sighting along the bottom edge at the horizon just below the Sun. Now mark the height of the Sun along the far vertical edge of the bag. Do this multiple times, and eventually the Sun won’t be climbing any more; the highest mark indicates the greatest height of the Sun, and you can draw the angle on the bag.
I don’t think we need that much accuracy. If we’re off by one minute for the sunrise time, that’s a maximum error of 16 miles or so (less away from the equator). Since the island is the only possible correct location within multiple miles, it wouldn’t take long to correct that kind of error.
The reason nobody suggested that is because it only works at the equinox. At any other time of year, that line will indeed curve. If I were to do it today, for instance, at noon the shadow would be north of the gnomon, and at sunrise and sunset, the shadow would be south of the gnomon. It’s tough to make a straight line do that.
I feel the need to point out how absurd this post is; the Truman show isn’t a real thing.
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