Powers and roots

problem
Rachel's problem

Age

14 to 16

Challenge level

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
problem
Deep roots

Age

14 to 16

Challenge level

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
problem
What an odd fact(or)

Age

11 to 14

Challenge level

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?
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
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 11

Challenge level

How many people were going to St Ives?
problem
Cubes within cubes

Age

7 to 14

Challenge level

We start with one yellow cube and build around it to make a 3x3x3 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?