# nice places on earth

We call a place on earth nice if you go 1 mile north, 1 mile west, 1 mile south, 1 mile east and then you end up exactly at the same place you started but you didn't visit any location more than twice. What is the minimum number of circumferences covering all nice places on earth?

• What is a “circumference”? Do you mean a great circle? Commented Oct 23, 2023 at 10:47
• if the points can be covered by one circle, they will count as one circumference Commented Oct 23, 2023 at 10:48
• Did you come up with this puzzle yourself? Commented Oct 23, 2023 at 13:43
• If I'm at the north pole, does standing still count as going 1 mile west or is it just impossible to go west from the north pole? Commented Oct 23, 2023 at 16:30
• @fblundun if you stand still you have not gone any distance in any direction. Whether or not you are at the north pole. Commented Oct 23, 2023 at 22:39

Two circles

The lines of latitude (east-west) are "parallel", but the lines of longitude (north-south) are not. Wherever you are, going 1 mile north and 1 mile south will take you to the same latitude. But going back east on a different line of latitude won't take you back to the same longitude anywhere, unless the east-west travel is the same distance north/south of the equator.

One circle is "parallel" with the equator and ½ mile south of it.

If you go west ½ mile north of the equator, you'll be going back east ½ mile south of the equator, which will take you back to the same latitude and longitude. You'll only be visiting the start/end point twice.

You can start the journey at any point on this circle.

Another circle is a bit over 1 mile south of the north pole, again along a line of latitude.

You start at $$A$$ and walk 1 mile north to $$B$$.
You walk west 1 mile, through $$C$$ and $$B$$ again to $$C$$.
You walk south 1 mile to $$D$$, and finally 1 mile east to $$A$$.
The path is shown in blue, with the duplicated section in magenta.

Let $$x$$ be the length of the path walked twice.
Let $$r$$ be the inner radius and $$R$$ be the outer radius.
Let $$\phi$$ be the angle between the longitude lines $$AN$$ and $$DN$$.

Assuming the earth's surface is locally flat, we known that
$$1 - x = 2 \pi r$$
$$R = 1 + r$$
$$\phi = \frac {x}{r} = \frac{1}{R}$$

From these we can derive
$$x^2 - 2x(1 + \pi) + 1 = 0$$
giving $$x = 0.123$$ miles or $$x = 8.161$$ miles.

I am unsure what use the larger value has, since it is more than 1 mile.
From the lower value of $$x$$ we can obtain
$$r = 0.1395$$ and $$R = 1.1395$$ and and $$\phi \approx 50^\text{o}$$.

So the second circle is $$1.1395$$ miles south of the North Pole.

The same could be used at the South Pole too, but you would begin on the smaller circle because you first need to walk north 1 mile. However as @fljx pointed out, you would visit the start/end point three times, so it's not a valid solution.

There may be other routes which involve more circuits around the North Pole (at a smaller radius), but they too will visit some locations more than twice.

• You may want to explain the (effect of the) but.. in the question. Commented Oct 23, 2023 at 12:43
• @Retudin I have extended the answer. Commented Oct 24, 2023 at 13:22
• Are you sure about your third circle? Don't you hit the start/end point three times? (At the start, while circumnavigating the pole eastward, and then at the end)
– fljx
Commented Oct 24, 2023 at 15:00
• @fljx thank you, I overlooked that. I've amended the answer. Commented Oct 24, 2023 at 15:26
• Thanks for the detailed solution. Commented Oct 24, 2023 at 17:03