07 Dec

A problem for the innocent minds

There is a well known problem of finding two points of the same color in the plane all points of which have been colored either red or blue distance 1 unit apart. It is simple but not trivial. This said what about finding two points of different colors under the extra condition that there are points of both colors.

So the problem is this:

The plane is colored in two colors: red and blue. Prove there are always two points of different colors 1 unit apart.

As usual, you are urged to give the problem a thought before looking at the solution.

The plane is colored in two colors: red and blue. Prove there are always two points of different colors 1 unit apart.

I'll give three solutions to the problem.

Solution 1

It's a condition of the problem that there is a pair of points of different colors. Consider any such pair. Draw a sequence of circles of radii 1, 2, 3, ... around each of the points.

Some of these circles are bound to intersect. Pick any point of intersection and join it to the two selected points by straight lines. On these lines mark the points at distance of 1 unit from each other. (These points are provisionally colored yellow.) Moving from, say, red point in steps of 1 unit, consider successive pairs of points. Since the entire stroll starts with a red point and ends with a blue point, the color had to have changed somewhere in-between. This gives you a pair of points 1 unit apart.

Solution 2

Join the two selected points by a straight line and start moving from the red point towards the blue one in steps of 1 unit. Color the provisional points yellow. There may be the case where the distance between the two points is integer. Keep this possibility in mind and consider the other case. On the latter occasion, we won't reach the blue point but get close enough, i.e., at the distance less than 1 unit. Form an isosceles triangle with sides 1 unit and the base joining the blue point to the last of the yellow points.

In both cases, like in Solution 1, we obtain a path from one of the selected points to the other that consists of unit steps. The same reasoning applies.

Solution 3

Consider the sets of all red points and all blue points. Both are not empty and their union covers the plane. There are border points - the points in whose vicinity, however small, there are points from two sets:

So there are always points of different colors at distance less than 1 unit. Join them by an isosceles triangle:

and apply the same reasoning as before.

Remark

There is a story behind the three solutions. When I first came across this problem, I immediately thought about the more popular and better known problem of finding two points of the same color. The problem is solved geometrically with circles and straight line segments. So, perhaps, the memory of that problem has conditioned me to think in these terms. The result was Solution 1 and Solution 2.

Now, last week, while driving my 8th grader to a swim practice, I posed him this problem. This is not the kind of the problem they work on in the honors geometry class. So he was stumped with it. To help him out I suggested that he should pick two points of different colors and consider three cases according as the distance between the points is equal to, less than, or more than 1 unit. He discharged the first case. I helped him with second one by pointing out the possibility of constructing an isosceles triangle. He then announced that this solves the problem because there are always points of different colors as close as we wish. He explained that this happens near any border point between the two monochromatic sets - essentially along the lines of Solution 3.

I was rather surprised that I have not thought about that at the outset. In fact I believe that this is the most natural solution to the problem. I shall never know whether I would have come up with it had it not been of my remembering the other problem. Had I really been conditioned?

Another question is bothering me: seeing how natural it was for my son to talk of sets and borders, may it be more appropriate for him to study elements of topology rather than the more traditional topics in elementary geometry?