Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
An article introducing continued fractions with some simple puzzles for the reader.