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# Visualising - Advanced

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Age 14 to 18

Challenge Level

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Age 14 to 18

Challenge Level

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Age 14 to 18

Challenge Level

How can visual patterns be used to prove sums of series?

Age 14 to 18

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

Age 14 to 18

Challenge Level

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

Age 16 to 18

Challenge Level

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

Age 16 to 18

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

Age 16 to 18

Challenge Level

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

Age 16 to 18

Challenge Level

How efficiently can various flat shapes be fitted together?

Age 16 to 18

Challenge Level

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

Age 16 to 18

Challenge Level

Explore the lattice and vector structure of this crystal.

Age 16 to 18

Challenge Level

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Age 16 to 18

Challenge Level

Use the diagram to investigate the classical Pythagorean means.

Age 16 to 18

Challenge Level

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

Age 16 to 18

Challenge Level

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

Age 16 to 18

Challenge Level

Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.

Age 16 to 18

Challenge Level

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

Age 16 to 18

Challenge Level

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

Age 16 to 18

Challenge Level

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?

Age 16 to 18

Challenge Level

In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

Age 16 to 18

Challenge Level

Can you find the link between these beautiful circle patterns and Farey Sequences?