Continued fractions

problem
All tangled up

Age

14 to 18

Challenge level

Can you tangle yourself up and reach any fraction?
problem
More twisting and turning

Age

11 to 16

Challenge level

It would be nice to have a strategy for disentangling any tangled ropes...
problem
Resistance

Age

16 to 18

Challenge level

Find the equation from which to calculate the resistance of an infinite network of resistances.
problem
Euclid's algorithm and musical intervals

Age

16 to 18

Challenge level

Use Euclid's algorithm to get a rational approximation to the number of major thirds in an octave.
problem
Golden fractions

Age

16 to 18

Challenge level

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
problem
Comparing continued fractions

Age

16 to 18

Challenge level

Which of these continued fractions is bigger and why?
problem
Not continued fractions

Age

14 to 18

Challenge level

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?
problem
There's a limit

Age

14 to 18

Challenge level

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
problem
Good approximations

Age

16 to 18

Challenge level

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.
article
Symmetric tangles

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!