Creating and manipulating expressions and formulae

problem
The medieval octagon

Age

14 to 16

Challenge level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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
Of all the areas

Age

14 to 16

Challenge level

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
problem
Seven up

Age

11 to 14

Challenge level

The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?
problem
Special sums and products

Age

11 to 14

Challenge level

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
problem
Sum equals product

Age

11 to 14

Challenge level

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?
problem
Three four five

Age

14 to 16

Challenge level

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
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
Hike and hitch

Age

14 to 16

Challenge level

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the road (at 4 miles per hour)...