Reasoning, convincing and proving

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

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?

Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?

Weekly Problem 31 - 2017
The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?

Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?

Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

If Clara spends £23 on books and magazines, how many of each does she buy?

Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?