Explaining, convincing and proving

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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?

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

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.