Explaining, convincing and proving

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Can you use the clues to complete these 3 by 3 Mathematical Sudokus?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Can you use the clues to complete these 6 by 6 Mathematical Sudokus?

Choose any three by three square of dates on a calendar page...

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?