According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have. . . .
What fractions can you find between the square roots of 65 and 67?
Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.
The challenge is to find the values of the variables if you are to solve this Sudoku.
Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
A bag contains 12 marbles. There are more red than green but green and blue together exceed the reds. The total of yellow and green marbles is more than the total of red and blue. How many of. . . .
Kyle and his teacher disagree about his test score - who is right?
Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?
Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?