I just discovered this topic with the posting of #4 and I have a geometric riddle I think in the spirit of the other fine problems. I devised the riddle below a few years ago and only shared it with my team (they were stumped) and a local MENSA bulletin (no feedback; maybe they were stumped, too).
The color-of-the-bear puzzle is a classic for kids. The puzzle, of course, is if you walk 1 mile toward the South from camp, then 1 mile East (or West), then 1 mile North and discover you’ve returned to camp, what color is the bear that has been ravaging your camp?
“White” since you are at the North pole.
Here’s the variation:
Forget the bear; it doesn’t matter. Where else in the world can you take the same path that returns to the starting point?
To help rein in the fussy, creative, over-thinkers, stick with the surface of a sphere that has a single N-S axis for the reference directions.
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Edit:
The “Almost Solution” award I’m giving to @ThomasU for being the first to get correctly half the complete answer without benefit of any other comments, being the only student (I think), and being in W. Linn where my brother lives!
[Award? I donated to that team’s GoFundMe. Best of luck at competitions.]
And once again ChiefDelphi shows it has really smart members! I’m impressed with what you did with my riddle.
There is a magnetic north and a true north. Since your puzzle didn’t clarify, I assume you were talking about the True North Pole. So the second location on Earth would be the Magnetic North Pole (the difference would be which “rules” you are following to walk the various directions, via compass or via rotational axis)
The next location I would propose is near (but not at) the center of the core of Earth. Moving “South” from your position represents the side of an equilateral triangle, then East/West represents the base, and “North” leads back to the starting point. I’d have to do some math to see where those North/South lines intersect the north and south poles.
FYI, you did say “IN the world”.
Nice, looks legit. But I think you still need to answer his question, where relative to the south pole is his starting point? It’s not 1 + 1/(2*pi) miles away
SPOILER ALERT! For anyone wanting to figure this out themselves do NOT read further!
Excellent Thinking! I never cranked the numbers and was pretty happy with the first sketch.
Here’s a big hint:
I can’t give full credit yet - maybe you get a grade C, B at best. (Take solace in the fact you’re the only one with any credit for an answer.) Keep up the good work. You can figure it out! I’ll accept Just a few words more for full credit; no math required to show you know the answer.
You got too complicated, the radius of the earth doesn’t matter (you can effective treat the earth as flat at these distances). The circumference is 1 mile, and we know the equation for the circumference, c=2rpi. That makes r 1/(2*pi), and the distance from the South Pole is 1+r.
Bonus: this also works for some fractional values of r as well. Divide r by 2 and you end up making 2 full circuits of the South Pole before returning to camp!
OK, but in an educational context, a point of the original “color of the bear” problem is to spark the learner’s thinking about non-euclidean geometries. If you flatten the earth then you remove this opportunity for exploration.
The original problem also doesn’t require any math at all to define the location. I think it’s always worth it to recognize the complexities of a problem, while also recognizing that, in a practical problem such as this there may be ways to simplify it to get close enough.
In his original solution,
s=0.1591549431349593939550202218306029548672483469701739332409428122
In the simplified,
r=0.1591549430918953357688837633725143620344596457404564487476673440
Thats a difference of .064 micrometers. For reference, the width of a human hair is 17-180 micrometers. As it pertains to the problem at hand, that’s functionally identical, and a heck of a lot easier to calculate.
YES! You got it!
There are an infinite number of locations as you approach the South Pole and make more and more complete circuits of the pole. You would be limited only by the size of your foot - ants could go around more times nearer to the pole, if they have a really warm coat so they don’t freeze to death.
Start anywhere on the set of increasingly close together circumferences laying in the spherical segment (mostly calculated above), go South, make an integral number of trips around the Earth, Head back North.
I conceived of this puzzle by thinking of the first 2-D sketch above and pushing can this be better and then figured the infinite solutions. I’ll encourage @Nate_Laverdure 's complication by saying I just learned something today by looking up what the bounding shape of the 3-D solution area is called - “Spherical Segment”
Sorry about upside down - I stole a sketch from the Internet and it is North oriented so I flipped it.
