Conjecturing and generalising

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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?

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.

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?