Conjecturing and generalising

Can you find sets of sloping lines that enclose a square?

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Delight your friends with this cunning trick! Can you explain how it works?