Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

An investigation that gives you the opportunity to make and justify predictions.

How many centimetres of rope will I need to make another mat just like the one I have here?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Find out what a "fault-free" rectangle is and try to make some of your own.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these three-digit numbers to the nearest 100?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

What happens when you round these numbers to the nearest whole number?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?