Explaining, convincing and proving

Show that it is rare for a ratio of ratios to be rational.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

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 the numbers from 1 to 1000 are written on a blackboard, which digit appears the most number of times?

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?

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