Explaining, 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?

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

If an athlete takes 10 minutes longer to walk, run and cycle three miles than he does to cycle all three miles, how long does it take him?

When is it impossible to make number sandwiches?

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

What can we say about all the primes which are greater than 3?

How many sets of three consecutive odd numbers can you find, in which all three numbers are prime?

Is there a quick and easy way to calculate the sum of the first 100 odd numbers?

Can you prove that the opposite angles of cyclic quadrilaterals add to $180^\circ$?