(Construction of regular n-sided polygons) The ancient Greeks solved the cases n = 3, 4, 5, 6. But for some n, such as n = 7, 9, the Greeks did not come up with a solution and this was left to later mathematicians. The famous mathematician Carl Friedrich Gauss (1777-1855) proved that construction of a regular n-sided polygon using compasses and straight edge is possible if and only if n = 2^mp₁p₂…p_{k, where m is a non-negative intger and} p₁, p₂, …, p_k are distinct Fermat primes. Refer to the notes Some Special Numbers for more information on Fermat primes.

(Trisecting the angle) To trisect any given angle. This has been proved to be impossible. However, owing to ignorance, many people continue to attempt attacking this problem. From time to time some people claim that they have solved the problem. Of course, all such ‘solutions’ are wrong, probably they have somewhere violated the basic postulates of construction. Incidentally, we are only allowed to use compasses and straight edge for finitely many times. Otherwise trisecting the angle would be possible.

(Doubling the cube) To construct the side of a cube with volume equal to twice that of a given cube. The impossibility of such construction is proved in many university textbooks in algebra.

(Squaring the circle) To construct a square with the same area as a given circle. This has also been proved to be impossible, and the crucial step was in 1882 when F. Lindemann (1852-1939) proved that p is transcandantal.

(Apollonius problem of tangent circles) Given three circles on the plane, construct a fourth circle tangent to these three circles.