The plane is divided into 3 disjoint sets. Can we always find two points in the same set a distance 1 apart? What about 9 disjoint sets?
Solution
Answer: yes, no.
Easy.
Suppose it is impossible. Take a triangle ABC with AB = AC = √3, BC = 1. A and B must be in the same set, because we can take two points X, Y such that AXY and BXY are equilateral triangles with side 1 (and then X and Y use up two sets, so A and B must both be the third set). Similarly A and C must be in the same set. But then B and C are in the same set and a distance 1 apart.
Tile the plane with squares side 0.7. Take all the points in a tile to be in the same set and label the tile 1 to 9 accordingly. Labels the tiles as follows:
...
...123123123...
...456456456...
...789789789...
...123123123...
...456456456...
...
Two points in the same tile are at most a distance 0.7 √2 < 1 apart, and two points in different tiles with the same color are at least a distance 2 x 0.7 = 1.4 > 1 apart.
© John Scholes
jscholes@kalva.demon.co.uk
12 Dec 1998