Question

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.

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Answer 1

Answer:

$\huge\bold\red{ANSWER}$

• Pick any two points A and B that have different colors. Starting at A , we can reach B using a finite number of steps, each of length exactly 1: just go directly towards B until the distance becomes less than 1, and at the end, if we didn't reach B exactly, we make two steps "to the side and back" to reach it. (Formally, if you are currently at C , imagine circles with radius 1 centered at B and C . Pick one of their two intersections, go from C to that intersection and from there to B .)

• As the first and the last point on this path have opposite colors, there has to be a pair of consecutive points with opposite colors, q.e.d.

• (Alternately, you could prove the new statement by contradiction. Pick any black point. All points in distance 1 from that point have to be black. This is the circle with radius 1. All points in distance 1 from those points have to be black as well. Here we can observe that the set of all points known to be black at this moment is the entire disc of radius 2 centered where we started. Continuing this argument, we can now grow the black disc indefinitely and thus prove that the entire plane has to be black, which is the contradiction we seek. Of course, this is basically the same proof as above, just seen from a different point of view.)

Answer 2

• picture which has black dots on a white background.
• picture which has black dots on a white background. This suggests that we color every point on the plane either white or black
• picture which has black dots on a white background. This suggests that we color every point on the plane either white or blackmaking sure to use infinitely many of each color.
• picture which has black dots on a white background. This suggests that we color every point on the plane either white or blackmaking 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.
• picture which has black dots on a white background. This suggests that we color every point on the plane either white or blackmaking 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.)

picture which has black dots on a white background. This suggests that we color every point on the plane either white or blackmaking 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.)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...

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