Powers and roots

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
More magic potting sheds

Age

11 to 16

Challenge level

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
problem
Magic potting sheds

Age

11 to 16

Challenge level

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
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
Pythagorean fibs

Age

16 to 18

Challenge level

What have Fibonacci numbers got to do with Pythagorean triples?
problem
Fibonacci fashion

Age

16 to 18

Challenge level

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
problem
Golden eggs

Age

16 to 18

Challenge level

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
problem
Lost in space

Age

14 to 16

Challenge level

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?
problem
Perfectly square

Age

14 to 16

Challenge level

The sums of the squares of three related numbers is also a perfect square - can you explain why?
problem
Guesswork

Age

14 to 16

Challenge level

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.