Reasoning, convincing and proving

This grid can be filled so that each of the numbers 1, 2, 3, 4, 5 appears just once in each row, column and diagonal. Which number goes in the centre square?

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Here's a chance to work with large numbers...

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

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

Can you work out where the blue-and-red brick roads end?