Conjecturing and generalising

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers and so on?

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

Here are two kinds of spirals for you to explore. What do you notice?

This task follows on from Build it Up and takes the ideas into three dimensions!

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.