Imagine you're standing on a set point on the surface on the outer coin, e.g. the one touching the inner coin. As the outer coin rotates around the inner coin, your experienced reality will be that you see n-1 rotations.

In the example of the outer coin having 1/3 the radius of the inner coin, as the outer coin rolls around the inner coin 4 times, you would actually only touch the inner coin 3 times.

I made a visualization for

> Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total?

https://www.shadertoy.com/view/7ddSR2

It matches experienced reality if you look at the stars rather than the sun, or if you use something like a Foucault pendulum to measure the rotation speed.

Why 367 dates?

Dates are not used to count Earth revolutions. They are counting days/nights and there are only a bit more that 365 of those in one year.