Eclipses. . .

When we experience a perfect, full solar eclipse here on Earth, the moon is directly between our line of sight and the sun. The result is that the sun is perfectly hidden by the moon, looking to us like a pitch black spot over the sun. Just a fraction of the sun's corona is visible around the outside edge of the moon. . . enough to really screw your eyes up if you look up at it with naked eyes sporting dilated pupils.

What are the odds that the Earth would come equipped with one (and only one) moon whose size and distance from earth make it perfect for eclipsing the sun for us?

Keep it the same size and move it farther from or closer to the Earth a significant amount, and you change how much of our view of the sky the moon occupies. This would change how much (or little) of the sun it blots out during a solar eclipse. Similarly, keep its distance from Earth the same but make it a bigger or smaller moon, and it stops being the perfect solar eclipse tool.

I know that the Earth's distance from the sun differs depending on its location along an elliptical orbit. And I know that the moon's distance from the earth fluctuates. Overall, though, I think the tolerances in those two variables are tight enough that full eclipses look very much the same here on planet Earth from year to year.

I wonder if there are any other planets in our solar system that have such a serendipitous combination of lunar and orbital factors?

Dunno. This is the kind of stuff that keeps me up at night.