Describing patterns and sequences

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?

MatildaMatildaMatil... What is the 1000th letter?

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

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

Explore one of these five pictures.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Find the next number in this pattern: 3, 7, 19, 55 ...

Investigate the successive areas of light blue in these diagrams.

Weekly Problem 38 - 2009
This sequence is given by the mean of the previous two terms. What is the fifth term in the sequence?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.