I have found recent puzzles here to be as delightfully counter-intuitive as others that deal with probability and infinity. They are the puzzles where each point on a line, a figure, and here the entire plane, receives a specific color. The notion of an isolated point (described by a pair of real numbers) is beyond common experience and sometimes requires proficiency in real number theory to work with. We may wonder what an isolated point actually is, and whether it has "adjacent" points. Nevertheless intuition can sometimes lead us to say, without rigor, that something must be or can't be true.

Example

For every 2-coloring of the plane (arbitrarily assign to every point (x, y) a red or blue color,) will there always be a unit equilateral triangle whose vertices are monochromatic?

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## bonanova

I have found recent puzzles here to be as delightfully counter-intuitive as others that deal with probability and infinity. They are the puzzles where each point on a line, a figure, and here the entire plane, receives a specific color. The notion of an isolated point (described by a pair of real numbers) is beyond common experience and sometimes requires proficiency in real number theory to work with. We may wonder what an isolated point actually is, and whether it has "adjacent" points. Nevertheless intuition can sometimes lead us to say, without rigor, that something must be or can't be true.

ExampleFor every 2-coloring of the plane (arbitrarily assign to every point (

,x) a red or blue color,) will there always be a unit equilateral triangle whose vertices are monochromatic?y## Link to comment

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