Reasoning, convincing and proving

problem
Ratty

Age

11 to 14

Challenge level

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
problem
Never prime

Age

14 to 16

Challenge level

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
problem
Napkin

Age

14 to 16

Challenge level

A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.
problem
Encircling

Age

14 to 16

Challenge level

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
problem
How many dice?

Age

11 to 14

Challenge level

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find there are 2 and only 2 different standard dice. Can you prove this ?
problem
Sitting pretty

Age

14 to 16

Challenge level

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
problem
Unit interval

Age

14 to 18

Challenge level

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
problem
Pareq exists

Age

14 to 16

Challenge level

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
problem
Chameleons

Age

11 to 14

Challenge level

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12 green, 15 brown and 18 yellow chameleons.
problem
The pillar of Chios

Age

14 to 16

Challenge level

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.