Help with a Martin Gardner maths riddle, please!

January 16, 2010 5:08 AM

Help with a Martin Gardner maths riddle, please!

Hello all. Am working my way through Martin Gardner's books of recreational maths in an effort to rekindle some sparks of thought in my ashen mind.

Just encountered one problem: an explorer walks one mile south, then one mile east, then one mile north, and finds himself back where he started. Where could he be, if not the North Pole?

I've solved that problem. But there is an addendum to the solution which goes on to say:

"Suppose we ignore the restrictions that the explorer walks south, east and then north. He walks a mile in any direction, turns 90 degrees, goes another mile, turns 90 degrees, walks a mile and finds himself back where he started. Where does he start? The answer, of course, is that he can start anywhere."

How (ahem) on earth could this be the case? I just don't get it. Help!

(NB - puzzle is from Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi by Martin Gardner)

posted by laumry to sports, hobbies, & recreation (20 answers total)

If he starts at the south pole, 'any direction' is always north.
So it's the same as the first case.
posted by hexatron at 5:20 AM on January 16, 2010

Hello Hexatron; thanks for the rapid answer. Certainly, the South Pole would work as one solution to the situation posited.

However, to be clear, my problem is the last sentence: "The answer, of course, is that he can start anywhere." I don't understand how the preceding text could be true at any point on the globe.
posted by laumry at 5:46 AM on January 16, 2010

Think about it this way: There's nothing special about the poles, they're just arbitrary points on a sphere. If you pick any other arbitrary point on the same sphere, you can treat it just like the poles. So you can walk in a (specific) direction, turn 90 degrees, etc. Think about rotating a globe so that your chosen point is at the top, then treat it like the North Pole.
posted by anaelith at 5:50 AM on January 16, 2010

I think I may have it...

laumry, congratulations, you now officially standing on the fnarth pole. The direction you start working is toward the sturth pole, since fnarth and sturth are opposite directions. You then turn 90 degrees and start heading shwest (or yeast), as yeast-shwest is perpendicular to fnarth-sturth. I believe a 90 degree turn will take you fnarth again to where you started.

I don't think that there's anything special about the north-south-east-west system, and that for any point on earth, you just make up a similar system with that as the pole, with the opposite pole in the direction you are walking.

Does that work?
posted by alphanerd at 5:58 AM on January 16, 2010 [1 favorite]

Bah!
posted by alphanerd at 5:58 AM on January 16, 2010

Great! Thanks everyone.
posted by laumry at 6:04 AM on January 16, 2010

What makes that statement perplexing is that it claims "turning 90 degrees" and "turning west" are equivalent. The problem is that our intuitive understanding of "turning 90 degrees" only applies to the plane, and that intuitive understanding does not extend to the sphere.

I think the point of the assertion is to point out this breakdown of our intuition, while at the same time giving us a place to start building our intuition on the sphere.

What we should be asking is, How do we measure angles on the sphere? Once we answer that question, the rest should be obvious.
posted by seliopou at 6:06 AM on January 16, 2010

I am a huge fan of Gardner, but I have to disagree with him here. Assuming by "walk one mile" he means "in a straight line," or more precisely "along a great circle," which is the nearest you can get to a straight line on the surface of a sphere. It should be pretty obvious to anyone who's ever lived in a city with a regular pattern that if you go one mile, turn 90 degrees, go one mile, turn 90 degrees again, and go one mile, you will not end up at the same place you started.

The problem is not with the phrase "turning 90 degrees." Angles are well-defined, even on the surface of a sphere, and north-south is perpendicular to east-west at all points except the poles where east-west is undefined (and in the original, neither of the two turns occur at the poles, so that's not the issue).

The reason you end up back at the original point is that lines of latitude—the route you trace when going due east or due west—are not great circles, except at the equator. Think of being near the north pole, and walking in a small circle around the pole with the pole on your left—you're heading due east the whole time, even though you're clearly not walking in a straight line.

In our daily experience we don't notice that due east and due west aren't straight lines, since we're far enough from the poles that they very nearly are. But travel due east in the northern hemisphere, and you will be curving ever so slightly to your left.

The effect becomes much more pronounced near the poles, where the east-west circles are much smaller. If you start at the pole, walk one mile south, then one mile east, then one mile north, the "one mile east" segment of your journey is pretty clearly not a straight line and will be noticeably curved.

It's true you could redefine your coordinate system so any place on earth would be a pole; but the analagous instructions would be, "walk one mile in any direction; turn 90 degrees and walk one mile along a circle whose center is at your original starting point; turn 90 degrees again (in the same direction as your first turn) and walk one mile. Then you'd end up at your starting point, but it's also clear that the second leg of your journey is not along a straight line [great circle].

A couple of observations: First, the second leg of the journey doesn't necessarily have to be the same distance as the first and third; it could be ten feet or ten miles and it would still work (making it the same just seems to make it more puzzling). Second, this works even on a plane; it has nothing to do with spherical geometry in particular, except inasmuch as we define north, south, east, and west on a sphere. We could even reconstruct it with the directional terms on a plane if we used polar coordinates, although the plane would have only a single pole, not the two poles of a sphere. Just define "north" as "towards the pole," south as "away from the pole," and east as "counterclockwise in a circle with the pole at the center," and the original instructions (start at the pole, go one mile south, one mile east, and one mile north) still return you to the pole, even though you're not on a sphere, and both of your turns were 90 degrees.
posted by DevilsAdvocate at 7:15 AM on January 16, 2010 [3 favorites]

Right; but I think Gardner's whole point was to make you realize that the "goes another mile" between the two 90º turns doesn't tell you to walk straight in either the polar or the general case — in the first it mean along along a line of latitude, and in the second it isn't specified. Your assumption that by 'walk one mile' he means 'in a straight line' is exactly the unwarranted assumption he wants to point out.
posted by nicwolff at 7:46 AM on January 16, 2010

But if you shouldn't assume "walk one mile" means "walk one mile in a straight line," why would you assume it means "walk one mile along a circle with your original starting point at the center," either? You might just as well take it to mean "walk one mile in a squiggly loopy path," and depending on where that ends up and which way you're facing, you may not end up back at the starting point. For that matter, if you shouldn't assume he means "in a straight line" for the second leg of the journey, why would you assume the first and third legs should be in a straight line?
posted by DevilsAdvocate at 8:10 AM on January 16, 2010

I understand the second problem. Just the first one is really puzzling me:
Just encountered one problem: an explorer walks one mile south, then one mile east, then one mile north, and finds himself back where he started. Where could he be, if not the North Pole?

Other than the North Pole I can only think of one 'special' point on the globe. And it can't be that one!
posted by 92_elements at 8:15 AM on January 16, 2010

I suppose even without any assumptions at all about whether each one-mile leg is straight or a circular arc or squiggly or anything else, it's possible to start at any point and end up in that same point, but that result seems rather trivial.
posted by DevilsAdvocate at 8:17 AM on January 16, 2010

92_elements: it doesn't work if you start at the other special point, but think about starting at a point near that other special point.
posted by DevilsAdvocate at 8:18 AM on January 16, 2010

For the second puzzle, I was all set to write a long comment, but the answer is "What DevilsAdvocate said". Due east isn't a straight line.

That being said, there's probably somewhere in Boston where you can walk a mile down some street, turn ninety degrees, walk another mile, turn ninety degrees, walk another mile, turn ninety degrees, and end up back where you started. This is because the streets there don't go straight.

92_elements: the answer to the first puzzle isn't the South Pole, but it's near the South Pole. There is a circle with center at the South Pole and circumference one mile; start one mile north of this circle.
posted by madcaptenor at 8:22 AM on January 16, 2010

Why assume the explorer is on earth?

The explorer could walk a mile in a straight line (great circle), turn any angle (not just 90 degrees) walk a mile in a straight line, etc. and end up in the same place.

...if the explorer was on an asteroid with a circumference of one mile (or 1/2 mile, or 1/3 mile...).
posted by cosmac at 8:32 AM on January 16, 2010

Also, a spherical triangle with sides one-fourth of the Earth's circumference can have three right angles. (Imagine a triangle with its corners at the North Pole and at two points on the Equator, ninety degrees apart.) So perhaps the explorer is on a spherical world with circumference four miles.

(Although why is there a spherical world of circumference four miles? Bunches of rock that small don't turn spherical under the influence of their own gravity.)
posted by madcaptenor at 8:38 AM on January 16, 2010

Also, a spherical triangle with sides one-fourth of the Earth's circumference can have three right angles.

I thought that any spherical triangle could have three right angles, regardless of size.
posted by 23skidoo at 9:53 AM on January 16, 2010

"regardless of size" should be "regardless of lengths of the sides"
posted by 23skidoo at 9:54 AM on January 16, 2010

What cosmac said, & madcaptenor, except for the "too small to form a sphere" bit. Just be on a small sphere. I'd say circumference of four miles, though.

Unless you're covering an eighth of the area of the sphere, 23skidoo, you can't quite do three right angles. If you approached sides of length zero, the angles would behave like those of a flat plane.
posted by Pronoiac at 12:21 PM on January 16, 2010 [1 favorite]

I remember the problem, but recall (perhaps incorrectly) that Gardner phrased it as 'turn East' and walk one mile, not 'turn 90 degrees'.
posted by dws at 1:08 PM on January 16, 2010