Explaining, convincing and proving

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

What is special about the difference between squares of numbers adjacent to multiples of three?

If you know the perimeter of a right angled triangle, what can you say about the area?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Can you explain what is going on in these puzzling number tricks?

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

Can you create a Latin Square from multiples of a six digit number?

A 2-digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.