I followed most of Alan Green’s ball-and-lamp illustration, but stumbled at the very end:

If you stand behind the ball and sight back to the light across that point, you will see that, from the Earth’s inhabitant’s point of view, the Sun is south of the line of latitude.

(Somehow I doubt buying him a soda would have been sufficient to lure him all the way over to the States with his props to give me the demonstration. I mean, I’d probably at least have to spring for a pint or two.)

But I think I finally got it when I drew myself a few simple pictures, which proved more precise for me than using an actual light and an actual ball. While I really liked your description of the difference between the Great Circle road and the Constant Latitude road, the sun appears to set even farther south than the latter. If I understand correctly, any time the South Pole is closer to the sun, the sun will seem to rise and set south of the observer’s latitude, whether the observer happens to be in the Northern or Southern Hemisphere (except near the poles, where the sun doesn’t even break the horizon). At the equinoxes, the poles are equidistant from the sun, and thus the sun rises and sets exactly due east and west, respectively.

http://solar.physics.montana.edu/YPOP/Classroom/Lessons/Sundials/winter.html

The sun actually sets just in west two days a year!

So the sun appears South during the “night”, and North during the “day”. Where you live gets a reduced form of this effect.

Your explanation includes the idea that great circles and latitude lines only coincide at the equator, but I find mine easier to think about.

I had another go at explaining with a ball and a lamp on my blog.

The mathematics matches reality and proves me wrong, but doesn’t satisfy my need for a rough understanding of how this paradox works (i.e. that an object in the North could appear in the South)

I just had a glimmer of understanding while in the shower. Let me see if I can convey it textually.

Imagine you have ordered a freshly terraformed small planet. Have the engineers make it a perfect sphere, covered in flat land.

Pick an arbitrary spot in the far south of the planet, and have the engineers build two roads intersecting at that point.

The first road should start describe a Great Circle around the planet. Its length will be equal to the circumference of the sphere. It should start out due West.

From the original vantage point, it will appear like a straight line, disappearing over the horizon. It doesn’t stick to due West. If you were to follow the straight line, you would notice your compass gradually turns closer to the North, then to the South. You will cross the equator twice.

Have the second road built to stick to the same latitude and head East-West around the planet.

How long is this second road? It is shorter than the first one - its length is proportional to the cosine of the latitude of the point you chose.

It will appear to curve to the left as you look West. The closer you are to the pole, the tighter the curve will be.

Some distance down the second road, have the engineers build a huge tower with a giant bright yellow glowing globe at the top.

Cover the second road in camouflage paint, so you can’t see it easily.

So standing at your arbitrary point, you can now see two landmarks.

One is the first road, that looks straight as a post, and - according to your compass - heads due West.

The other is a giant yellow globe. It appears to the left (South) of the first road, and yet it is exactly due West of you.

Is this the visual solution to the paradox? Does this adequately explain how an object, such as the Sun, can appear further South than you (i.e. left of West) even if it isn’t?

This FAQ recommends Astronomical Algorithms by Jean Meeus.

[Ed: Broken link to PyEphem corrected.]