Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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?
Balance the bar with the three weight on the inside.
Attach weights of 1, 2, 4, and 8 units to the four attachment
points on the bar. Move the bar from side to side until you find a
balance point. Is it possible to predict that position?
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.
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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
Which rational numbers cannot be written in the form x + 1/(y +
1/z) where x, y and z are integers?
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. . . .
What fractions can you find between the square roots of 65 and 67?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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. . . .
Christmas trees are planted in a rectangular array of 10 rows and
12 columns. The farmer chooses the shortest tree in each of the
columns... the tallest tree from each of the rows ... Which is. . . .
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. . . .