What happens when a procedure calls itself?
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
It's like 'Peaches Today, Peaches Tomorrow' but interestingly generalized.
Keep constructing triangles in the incircle of the previous triangle. What happens?
Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?
How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
This problem explores the biology behind Rudolph's glowing red nose, and introduces the real life phenomena of bacterial quorum sensing.
Form a sequence of vectors by multiplying each vector (using vector products) by a constant vector to get the next one in the seuence(like a GP). What happens?
Can you find the link between these beautiful circle patterns and Farey Sequences?
