1. Russian King Peter the Great announced that no member of the Royal Family should marry until they pass in arithmetic, geometry and navigation.
2. There is no equilateral triangle, the coordinates of whose vertices are all integers.
3. There are only five Heronian triangles (triangles having integer sides) such that the perimeter is equal to the area. They are: (5, 12, 13) , (6, 8, 10) , (6, 25, 29) , (7, 15, 20) and (9, 10, 17). The first two triangles are right triangles also.
4. A plane can be filled with squares, equilateral triangles and regular hexagons. No other regular polygons can fill the plane without gaps.
5. Three legged stool is more stable than any other stool. The principle underlying is this fact is we can draw only one circle through three given points which are not collinear. That is why old people walk with the assistance of a stick (third leg) , or even think of tripod.
6. An angle of 3 degrees is the only constructible angle of prime measure.
7. There are certainly at least two peoples in the world with exactly the same number of hairs on their heads! (There are more people than the hairs on any one head).