Question

Consider a square whose side-length is one unit. Select any five points from inside this square. Prove that at least two of these points are within \(\frac{\sqrt{2}}{2}\) units of each other.

Short answer

By the Pigeonhole Principle, when dividing the unit square into four smaller squares, at least two out of five points must be in the same smaller square. The maximum possible distance between two points in one of these smaller squares can be calculated using the Pythagorean theorem and is found to be \(\frac{\sqrt{2}}{2}\) units, proving the proposition.

Step-by-step solution

Divide the Square

Divide the square with side length of one unit into four equal squares. Each smaller square will have a side-length of \(\frac{1}{2}\) units.

Place the Points

Place five points inside the larger square. According to the Pigeonhole Principle, since there are more points than smaller squares, at least two points must be placed in the same smaller square.

Check Distance between Points

By the Pythagorean theorem, the distance between any two points within one of the smaller squares (which is a right triangle with sides of length \(\frac{1}{2}\)) is less than or equal to the length of the hypotenuse, which is \(\sqrt {(0.5)^2 + (0.5)^2}\) or \(\frac{\sqrt{2}}{2}\) units.