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

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

Benâ€™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?

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?

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

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

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

Got It game for an adult and child. How can you play so that you know you will always win?

Can you find all the ways to get 15 at the top of this triangle of numbers?

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.

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?

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?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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.

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

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?

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

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

Are these statements always true, sometimes true or never true?

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

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

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

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

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

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?

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?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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

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

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?

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

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

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

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?

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

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

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

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?

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