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?

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

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

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?

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

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

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

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?

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

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

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

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?

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

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?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

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

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

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

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?

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

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?

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?

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.

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

Can you explain the strategy for winning this game with any target?

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?

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

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

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

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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

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

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

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?

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?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

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

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?

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

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

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

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