Explaining, convincing and proving

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

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

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Here's a very elementary code that requires young children to read a table, and look for similarities and differences.

Max and Bryony both have a box of sweets. What do you know about the number of sweets they each have?

How many rectangles can you see? Are they all the same size? Can you predict how many rectangles there will be in counting sticks of different lengths?