This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

Which of these triangular jigsaws are impossible to finish?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

How many different colours of paint would be needed to paint these pictures by numbers?

This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box.

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?