The Art of Deduction

Creating convincing arguments or "proofs", to show that statements are always true, is a key mathematical skill. The problems in this feature offer you the chance to explore geometrical properties, make conjectures and create proofs to show that these are always true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs.  

Plus magazine has a selection of interesting articles exploring proofs in which pictures play an important role.

Pentakite 

Age 14 to 18
Challenge Level
Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Quad in Quad 

Age 14 to 18
Challenge Level
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Kite in a Square 

Age 14 to 18
Challenge Level
Can you make sense of the three methods to work out what fraction of the total area is shaded?

The Converse of Pythagoras 

Age 14 to 18
Challenge Level
Can you prove that triangles are right-angled when $a^2+b^2=c^2$?

To Swim or to Run? 

Age 16 to 18
Challenge Level
The famous film star Birkhoff Maclane wants to reach her refreshing drink. Should she run around the pool or swim across?


We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.