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Multi-Color Rectangles

Consider a 3x3 array of dots. Color each of the 9 dots, such that any 4
dots at the corners of a rectangle are not all the same color. Rectangles
may be found at any angle. A 3x3 array has 10 rectangles. What is
the minimum number of colors needed? For example, in a 2x2 array, with
only one rectangle, 2 colors are needed to make the rectangle use more than one
color.
- What is the maximum number of dots in the array that can be colored
green?
- How many colors would be needed if every rectangle must have four
different colors at its corners?
- Repeat for a 4x4 array of dots.
- Repeat for a 5x5 array of dots.

Source: Original, though I've seen similar before.

Solution

Mail to Ken