Describing patterns and sequences

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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?

There are lots of ideas to explore in these sequences of ordered fractions.

What are the last two digits of 2^(2^2003)?

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?

This number has 903 digits. What is the sum of all 903 digits?

What is the last digit of the number 1 / 5^903 ?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

How many different sets of numbers with at least four members can you find in the numbers in this box?