Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Explain how to construct a regular pentagon accurately using a straight edge and compass.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.