Conjecturing and generalising

problem
Great Granddad

Age

11 to 14

Challenge level

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?
problem
Blue and white

Age

11 to 14

Challenge level

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
problem
One o five

Age

11 to 14

Challenge level

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by 3, 5 and by 7...
problem
Chocolate maths

Age

11 to 14

Challenge level

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it works?
problem
Converging means

Age

14 to 16

Challenge level

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the two sequences.
problem
Gnomon dimensions

Age

14 to 16

Challenge level

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
problem
Which scripts?

Age

7 to 11

Challenge level

There are six numbers written in five different scripts. Can you sort out which is which?
problem
Parabolic patterns

Age

14 to 18

Challenge level

The illustration shows the graphs of fifteen functions. Two of them have equations $y=x^2$ and $y=-(x-4)^2$. Find the equations of all the other graphs.
problem
AMGM

Age

14 to 16

Challenge level

Can you use the diagram to prove the AM-GM inequality?
problem
Enclosing squares

Age

11 to 14

Challenge level

Can you find sets of sloping lines that enclose a square?