Diagram for problem 18 |

__Problem 18__: The points of a plane are colored with three colors. Show that there exist two points with distance 1 both having the same color.

__Solution__: (With reference to the diagram)

Distance between centres of 2 circles on
the left and right, i.e. between the 2 points coloured a, is
√3. Measure the same distance from the left a and find the corresponding point of
intersection on the right circle, i.e. the point on the latter circle which is
at a distance of √3 from the left a. Suppose that point is coloured b.
Draw a unit circle centred at that point. The points where this new circle
intersects circle 1 must be coloured c since points on circle 1 must be of
colour b or c.
Besides, these 2 points of intersection must be at a distance of 1 unit from
each other since the centres of circles 1 and 3 are √3 units apart.

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