I came across this problem of partitioning 3 space into circles (https://www.cut-the-knot.org/proofs/tessellation.shtml)

For the life of me I cannot understand the solution at the end of that page :(

I cannot see how each of the red spheres has only 2 points of intersection with a black circle. For eg consider the red sphere with radius 1. Surely it intersects the whole left half of the black circle centered at x = 1.

I would greatly appreciate if someone could point me on the right path to understanding this.

Thanks :)

The sphere in your example is

x² + y² + z² = 1

while the circle is

(x - 1)² + y² = 1, z = 0

Eliminating y and z gives

x² = (x - 1)²

from which x = 1/2. Then, for y, there is a quadratic equation:

y² = 3/4.