Pythagoras' theorem

problem

Circle Packing

Age

14 to 16

Challenge level

Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...
problem

Round and Round

Age

14 to 16

Challenge level

Prove that the shaded area of the semicircle is equal to the area of the inner circle.
problem

Grid Lockout

Age

14 to 16

Challenge level

What remainders do you get when square numbers are divided by 4?
problem

Rhombus in Rectangle

Age

14 to 16

Challenge level

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
problem

Matter of Scale

Age

14 to 16

Challenge level

Can you prove Pythagoras' Theorem using enlargements and scale factors?
problem

Corridors

Age

14 to 16

Challenge level

A 10×10×10 cube is made from 27 2×2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
problem

Circle Box

Age

14 to 16

Challenge level

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
problem

Folded Over

Age

14 to 16

Challenge level

A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?
problem

Walk the Plank

Age

14 to 16

Challenge level

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?
problem

The Fire-Fighter's Car Keys

Age

14 to 16

Challenge level

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket?