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.

Find the sum of all three-digit numbers each of whose digits is odd.

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?

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

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 ...?

Try out this number trick. What happens with different starting numbers? What do you notice?

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

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

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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.

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?

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?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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

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

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?

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?

This task follows on from Build it Up and takes the ideas into three dimensions!

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

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?

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.

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?

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?

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

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?

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

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Here are two kinds of spirals for you to explore. What do you notice?

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

This challenge asks you to imagine a snake coiling on itself.

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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?

This activity involves rounding four-digit numbers to the nearest thousand.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?