Question

1. There are infinite black and white dots on a plane. Prove that the distance between one black dot and one white dot is one unit.

2. How can you drop two eggs the fewest amount of times, without them breaking?

Answer 1

If you click on the link you find a picture which has black dots on a white background. This suggests that we color every point on the plane either white or black, making sure to use infinitely many of each color. With these constraints it is indeed the case that there must be a black point and a white point at unit distance. (In fact, "infinite" can be weakened to nonempty.)

Explanation:

Hint: starting with a black point, we're done unless the entire unit circle around that point is black. Now repeat that argument for each point on that unit circle: we've already generated a sizable swathe of the plane colored totally black...