Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), where squares are used the same amount of times in the design of the figure.
Consider a segment on a line of length \(3m\). Jack chose \(4\) random points on the segment and measured all the distances between points. Prove that at least one of the distances is less or equal than \(1m\).
The kingdom of Triangland is an equilateral triangle of side \(10\) km. There are \(5\) cities in this kingdom. Show that some two of them are closer than \(5\) km apart.
Margaret marked three points with integer coordinates on a number line with red crayon. Meanwhile Angelina marked a midpoint for every pair of red points with blue crayon. Prove that at least 1 of the blue points has an integer coordinate.
Margaret and Angelina decided to move to the second dimension. Now Margaret marked five points with both integer coordinates on a plane with red crayon. While Angelina marked a midpoint for every pair of red points with blue crayon. Prove that at least 1 of the blue points has both integer coordinates.
Anna has a garden of square shape with side \(4\) m. After playing with her dog in the garden she left \(5\) dog toys on the lawn. Show that some two of them are closer than \(3\) m apart.
Twelve lines are drawn on the plane, passing through a point \(A\). Prove that there are two of them with angle less than \(17^{\circ}\) between them.
Inside a square of area \(6\), there are three polygons, each of area \(3\). Show that some two of them overlap and the area of the overlap is at least \(1\).
A Wimbledon doubles court is 78 ft \(\times\) 36 ft. After a long practice match, there were 79 tennis balls in the court area. Show that some two of them were no further than \(6\sqrt{2}\) ft away.
There are \(n\) ambassadors sitting at a round table. Each ambassador has a flag of a country in front of them. Unfortunately flags have been mixed up and as a result each ambassador has a wrong flag in front of them, while their flag is in front of some other ambassador. Show that you can rotate the table with the flags on it, in such a way that at least two ambassadors will have correct flags in front of them.