Puzzles

Pick two points (A and B) of the same colour (say red). Consider the points:

If at least one of C, D or E is red, then three equally spaced red points have been found. If C, D and E are blue, then they are three equally spaced blue points.

Will there be four equally spaced points of the same colour?

Yes. To show this, let's suppose there is an arrangement of pins with no monochrome triangle.

Let the middle pin be blue (if it is not blue, then swapping all the colours will make it blue without adding/removing monochrome triangles).

Pins 8, 2 and 6 make a triangle. Therefore one of these pins must be blue. By symmetry, any of these situations is equivalent to 8 being blue.

Pins 9 and 4 must be red to prevent a blue triangle.

Pin 3 must be blue to prevent a red triangle.

Pin 6 must be red to prevent a blue triangle.

But now whichever colour 10 is, it will make a monochrome triangle. Therefore there must be three pins of the same colour which lie on the vertices of an equilateral triangle.

What is the smallest \(n\) such that any mixture of \(n^2\) red and blue pins arranged in a square grid must contain four pins of the same colour which lie on the vertices of a square?

It can be done with two colours. Let's call these red and blue.

Draw the polygon and colour it red. As each line is added to is, swap the colours on one side of the line which leaving them the same on the other side. At the end of this process you will have any polygon coloured with two colours.

How many regions will the regular shape be split into by the lines?