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Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Always Perfect

Age 14 to 18
Challenge Level

Part 1

Have you tried some simple cases to see if there is a pattern?

Can you adapt Charlie's and Claire's methods to prove your conjectures for parts (b), (c) and (d)?

Part 2

How could you write four consecutive numbers?

Can you use a table/grid to help you factorise your expression?

In the video below, Claire shows how you could factorise a quartic expression.