Powers and roots

problem

Lastly - Well

Age

11 to 14

Challenge level

What are the last two digits of 2^(2^2003)?
problem

Plus or Minus

Age

16 to 18

Challenge level

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
problem
Favourite

Equal Temperament

Age

14 to 16

Challenge level

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.
problem

Bina-Ring

Age

16 to 18

Challenge level

Investigate powers of numbers of the form (1 + sqrt 2).
problem

Integer Indices

Age

14 to 16

Challenge level

From this sum of powers, can you find the sum of the indices?
problem

St Ives

Age

7 to 14

Challenge level

How many were going to St Ives?
problem
Favourite

Cubes Within Cubes

Age

7 to 14

Challenge level

We start with one yellow cube and build around it to make a 3×3×3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
problem
Favourite

Pocket Money

Age

11 to 14

Challenge level

Which of these pocket money systems would you rather have?
problem

Largest Number

Age

11 to 14

Challenge level

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.
problem
Favourite

Power Mad!

Age

11 to 14

Challenge level

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.