Conjecturing and generalising

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Norrie sees two lights flash at the same time, then one of them flashes every 4th second, and the other flashes every 5th second. How many times do they flash together during a whole minute?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Using the picture of the fraction wall, can you find equivalent fractions?