Conjecturing and generalising

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Explore the effect of combining enlargements.

Delight your friends with this cunning trick! Can you explain how it works?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Can you find the values at the vertices when you know the values on the edges?

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?