Sunday, January 13, 2013

Colouring problems from Arthur Engel's book - problem 18

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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